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FOUNDED BY JOHN D. ROCKEFELLER 



THE SIGNIFICANCE OF THE MATHE- 
MATICAL ELEMENT IN THE 
PHILOSOPHY OF PLATO 



A DISSERTATION 

SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF ARTS 

AND LITERATURES IN CANDIDACY FOR THE DEGREE OF 

DOCTOR OF PHILOSOPHY 

(department of philosophy) 



BY 



IRVING ELGAR MILLER 




CHICAGO 
THE UNIVERSITY OF CHICAGO PRESS 

1904 



Gbe THnft>ersft$ of Cbfcago 

FOUNDED BY JOHN D. ROCKEFELLER 



THE SIGNIFICANCE OF THE MATHE- 
MATICAL ELEMENT IN THE 
PHILOSOPHY OF PLATO . 



A DISSERTATION 

SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF ARTS 

AND LITERATURES IN CANDIDACY FOR THE DEGREE OF 

DOCTOR OF PHILOSOPHY 

(DEPARTMENT OF philosophy) 



BY 

IRVING ELGAR MILLER 



CHICAGO 
THE UNIVERSITY OF CHICAGO PRESS 

1904 






Copyright, 1904 
By the University of Chicago 

mm 

$2705 



February, 1901 



TABLE OF CONTENTS. 



Chapter I. Plato's General Attitude toward Mathematics. 
Plato an admirer of mathematics . 
Interest in its qualities and characteristics 
Disciplinary value 
Clearness and certainty 
Intuitive element 



Conception of definition 

Method of procedure . 

Necessity and universality 

Utility of mathematics 

Scientific aspect . 

Greek ignorance of mathematics 

Influence on mathematics of Plato's philosophic interest 

Chapter II. The Formulation of Philosophical Problems. 
Significance of unconscious factors .... 

Early Greek philosophy 

Protagoreanism . 

The Socratic factor 

Ethical point of departure . 

Reaction against Protagoreanism 

Distinction between senses and intellect 

Relation of mathematics to this distinction . 

Analogy of the arts . . . . . 

This analogy and the problem of ethics .... 

Analogy of the arts in its relation to the ontological problem 
Mathematics and ontology ...... 

Return to the epistemological problem .... 

Bearing on the ethical problem ..... 

Further development of the analogy of the arts in relation to 
problem ........ 

Ethical problem limited by epistemological . 

Influence of mathematics upon Plato's conception of the 

Mathematics and cosmology ..... 

Relation of mathematics to idealism .... 

The figure of the divided line, in particular . 

Chapter III. Method, or the Technique of Investigation. 
Plato's interest in method ...... 

His dogmatism and its relation to method 

Mathematics and method ...... 

Socratic and Platonic attitudes compared 
Relation of mathematics to the Platonic attitude . 



the 



ethical 



9 

9 

10 

10 
10 

II 
II 

12 
12 
15 

16 

17 

21 
21 
22 
23 

24 
24 

25 
25 
28 
28 

29 
30 
30 
3i 

33 
33 
34 
37 
40 
44 

47 
50 
52 
52 
53 



4 THE MATHEMATICAL ELEMENT IN PLATO S PHILOSOPHY 

PAGE 

Mathematics and the Socratic universals . . . . . -54 

Mathematics and the intuitive element (the doctrine of recollection) . 54 
Mathematics and definition ......... 55 

The Method of Analysis : 
The method in mathematics ......... 57 

Its positive phase .......... 58 

Its negative phase .......... 58 

Socratic analysis .......... 59 

Socratic analysis as seen in the minor dialogues . . . . .59 

Criticism of Socratic analysis . . . . . . . .61 

Zeno's analysis ........... 62 

Platonic Analysis : 

The Gorgias ........... 63 

The Meno ............ 65 

The Euthydemus .......... 66 

The Republic 67 

The Phcedo 68 

The Thecetetus ........... 69 

The Parmenides .......... 72 

The Sophist ........... 74 

The Statesman ........... 76 

Summary ............ 76 

Relation of mathematics to Platonic analysis . . . . . .77 

Chapter IV. The Relation of Mathematical Procedure to Dialectic. 

Logical statement of the ethical problem in its relation to demand for 

method ........... 79 

Mathematical method suggestive as to philosophic method . . .82 

Dialectic distinguished from eristic ....... 82 

Dialectic and mathematical method compared . . . . .84 

The nature of dialectic, and its relation to the solution of the epistemo- 

logical and ethical problems ........ 86 

Return to comparison of mathematical method and dialectic . . .87 

The character of the distinctions which Plato sets up . . . .90 

Final solution of the ontological and the cosmological problems . .91 

Bibliography. 



TABLE OF REFERENCES 

TO PASSAGES IN PLATO INVOLVING MATHEMATICS. 

Based on The Dialogues of Plato, translated by B. Jowett, M.A., 
in five volumes, third edition. (London : Oxford University Press, 
1892; New York: The Macmillan Co., 1892.) 



DIALOGUE AND 
MARGINAL PAGE. 


VOLUME AND PAGE 
IN JOWETT. 


DIALOGUE AND 
MARGINAL PAGE. 


VOLUME AND PAGE 
IN JOWETT. 


Protagoras, 356~57 


I, l8l 


Timaeus, 43 . . . 


HI, 463 


Euthydemus, 290 


I, 227 


Parmenides, 143-44 


IV, 68-71 


Cratylus, 436 


I, 384 


Theaetetus, 147-48 . 


IV, 199-200 


Phaedrus, 274 




I, 484 


Theaetetus, 162 . . 


IV, 218 


Meno, 81-86 . 




II, 41-47 


Theaetetus, 185 . . 


IV, 246 


Meno, 86, 87 




II, 48 


Theaetetus, 198-99 . 


IV, 263-64 


Euthyphro, 12 




II, 88 


Statesman, 257 . . 


IV, 451 


Phaedo, 92 . 




II, 237 


Statesman, 258-60 . 


iv, 452-55 


Phaedo, 96-97 




II, 242-43 


Statesman, 266 . . 


IV, 462-63 


Phaedo, 101 . 




II, 247 


Statesman, 283-85 . 


IV, 483-86 


Phaedo, 104 . 




II, 250-51 


Philebus, 24-25 . . 


IV, 590-91 


Phaedo, 106 . 




II, 253-54 


Philebus, 51-52 


IV, 625^26 


Gorgias, 450-51 




II, 329-30 


Philebus, 55~58 . . 


IV, 630-33 


I. Alcibiades 126 


II, 494 


Philebus, 64-65 . . 


IV, 641-43 


Republic, 5 = 458 


. . HI, 152 


Laws, 4 : 717 . . . 


V, 100 


Republic, 6:510-11 


III, 211-13 


Laws, 5:737-38 . 


V, 119-20 


Republic, 7:521-34 


. Ill, 221-38 


Laws, 5:746-47 • 


V, 128 


Republic, 8:545"47 


. Ill, 249-51 


Laws, 5:746-47 


V, 129-30 


Republic, 9:587-88 


. Ill, 300-301 


Laws, 6 : 771 . . . 


V, 152-53 


Republic, 10:602-3 


III, 316-17 


Laws, 7:809 . . . 


V, 191 


Timaeus, 31-32 . 


. • 111,451 


Laws, 7:817-822 


V, 200-206 


Timaeus, 38-39 




. • 111,457-58 


Laws, 9 : 877 . . . 


V, 262 



INTRODUCTION. 

Plato took a deep interest in mathematics ; philosophy was his 
passion. These two interests, at first thought disparate, came into 
a relation of thoroughgoing intellectual interaction. Plato's mathe- 
matical studies had a different motive, aspect, and outcome from the 
fact that he was primarily a philosopher ; his philosophy had a 
different quale, from the fact that he was a devotee of mathematics. 

It was significant for the progress of mathematics that when 
Plato turned his attention toward this science he looked with the 
eyes of a philosopher. Hence I shall discuss what it was that his 
philosophic insight saw in mathematics to attract him, and in what 
way the philosophic attitude of mind which he brought to bear on 
the study of this subject served to further the progress of the science. 

On the other hand, the interaction of the mathematical and the 
philosophic elements was an important factor in the development 
of Plato's philosophic system. The main part of this book will be 
given up to the task of showing the influence of mathematics upon 
the formulation of philosophic problems, in the determination of 
method, and as affecting the content of philosophy. 

In the first chapter I have put the mathematical element in the 
foreground wth special reference to showing the significance to 
mathematics of the philosophical element. In the remaining chapters 
I have put the philosophical element into the foreground and have 
sought to show the influence upon it of the mathematical element. 
This has involved a duplication in the treatment of certain topics 
and considerable cross-reference at certain points. This element of 
repetition might have been avoided by a unification of treatment 
under the lead of the philosophical aspect, with the mathematical as 
incidental and subsidiary. But I have thought that the advantages of 
giving the mathematical element a more unified discussion on its own 
account counterbalanced the disadvantages from the other point of 
view. 

No attempt has been made to deal with the so-called number 
theory of the Pythagoreans, into harmony with which it is some- 
times said that Plato cast his philosophy later in life. The authority 
for setting up this relationship between mathematics and Plato's 

7 



8 THE MATHEMATICAL ELEMENT IN PLATO'S PHILOSOPHY 

philosophy is very problematic, to say the most. It finds very little, 
if any, support in Plato's own writings. Again, the reader who is 
looking for solutions of the mathematical puzzles to be found in the 
dialogues will look in vain, except as some of these puzzles may 
find rational explanation from the point of view which is developed 
in this book — a point of view which is concerned with the move- 
ment of thought, and hence views the introduction of mathematical 
ideas, not alone from the side of their intrinsic character or worth, 
but primarily with reference to their bearing upon the philosophical 
problems in relation to which they stand. 



CHAPTER I. 

PLATO'S GENERAL ATTITUDE TOWARD MATHEMATICS. 

The dialogues of Plato abound in allusions and references to 
mathematics. It is not difficult to see that he is a great admirer of 
the mathematical sciences and has a keen appreciation of their value. 
Let us take up a little evidence of a general order before proceeding 
to details. 

Mathematical study fascinates Plato by reason of its " charm." x 
It is through this quality that solid geometry is enabled to make 
progress, even though it is as yet undeveloped, generally unappre- 
ciated, and poorly taught, 2 Arithmetic is declared to have a " great 
and elevating effect." 3 It is a " kind of knowledge in which the best 
natures should be trained," 4 being an essential to manhood. 5 This 
latter conception of the value of mathematics is asserted very strongly 
in what is probably the very latest of Plato's dialogues — the Laws. 
There he argues that " ignorance of what is necessary for mankind 
and what is the proof is disgraceful to everyone." Some degree of 
mathematical knowledge is " necessary for him who is to be reckoned 
a god, demigod, or hero, or to him who intends to know anything 
about the highest kinds of knowledge." 6 "To be ignorant of the 
elementary applications of mathematics is ludicrous and disgraceful, 
more characteristic of pigs than of men." 7 

In such high terms Plato expresses his appreciation and admira- 
tion of the mathematical sciences. Further and more detailed 
investigation will show more specifically the nature of his attitude 
toward this subject and the grounds upon which it rests. His 
estimate of the value of mathematical study grows out of a philo- 
sophical attitude of mind rather than a practical one. What his 
attitude toward utility was will be taken up in detail later. Suffice 
it to say here that his main interest in mathematics centered in its 
qualities, characteristics, the mental processes and methods involved, 
the possibilties which he saw in it of scientific procedure, and the 
suggestions and analogies which it furnished him in the field of 
philosophic processes, methods, and results. 

1 Rep., 7 : 525. 3 Rep., 7 ■ 5^5- 5 Rep., 7 '• 522. 

2 Rep., 7 : 528. i Rep., 7 : 526. 6 Laws, 7:818. 7 Laws, 7 : 819. 



10 THE MATHEMATICAL ELEMENT IN PLATO S PHILOSOPHY 

Of the qualities and characteristics of mathematical study which 
Plato regarded as valuable, one of the most important is its general 
disciplinary value. Anyone interested so much as he was in the 
cultivation of the reasoning processes could not help seeing the 
possibilities of mathematics in this respect, even fragmentary as the 
science was in his day. He observes that mathematical training 
makes one, even though otherwise dull, much quicker of apprehen- 
sion in all other departments of knowledge than one who has not 
received such training. 8 So much is he impressed with this fact that 
when he has once made the point in his discussion of arithmetic, 9 he 
repeats it in his discussion of geometry. 10 It is on account of the 
training which mathematics gives in the power of abstraction and in 
reasoning processes, aside from its idealistic tendency (to be dis- 
cussed later), that Plato makes mathematical study a propaedeutic to 
philosophy. 11 

Though all sciences in the time of Plato were in a more or less 
embryonic stage of development, mathematics among them, yet this 
subject, by reason of the comparative simplicity of its elements, had 
advanced farther than the rest and stood out as rather conspicuous 
for the clearness of its procedure and the certainty of its results. 
Such a fact as this is of more interest to Plato than any utilitarian 
value that may arise from the exactness of mathematics. He has 
a philosophic appreciation of the fact that the arts which involve 
arithmetic and the kindred arts of weighing and measuring are the 
most exact, and of these those " arts or sciences which are animated 
by the pure philosophic impulse [i. e., theoretical or pure mathe- 
matics] are infinitely superior in accuracy and truth." 12 The reason 
for this clearness and certainty was felt to lie in three important 
features: (i) the intuitive element in mathematics, (2) its more 
correct conception of definition, and (3) its method of procedure. 
As these points come up for further discussion later on in another 
relation, only the briefest elaboration of them will be undertaken 
here. 

That Plato was impressed by the intuitive element in mathe- 
matics is certain from the reference in the Meno, if we had no other. 
When he wants an illustration of his doctrine of knowledge as 
recognition of that which was perceived in a state of being ante- 

8 Cf. Laws, 5 : 747. 

8 Rep., 7:526. "Rep., 7:521-33; see especially 533. 

" Rep., 7 : 527. u Phileb., 55-57- 



PLATO S GENERAL ATTITUDE TOWARD MATHEMATICS 1 1 

cedent to this life, he turns to mathematics. The slave boy in the 
Meno is made to go through a demonstration in geometry where 
" without teaching," but by a process of questioning, he recovers his 
knowledge for himself. 18 Whether Plato felt the full force of its 
significance or not, what he really brought out in this practical illus- 
tration was the intuitive element in mathematics. Is it unreasonable 
to think that it was this intuitive element in mathematics which either 
created or was a factor in creating the philosophical problem the 
solution of which Plato sought in his doctrine of recollection ? 

Mathematics had, to a higher degree than other subjects, also 
attained to a correct conception of definition. That one of the 
reasons for Plato's appreciation of mathematics is to be found in this 
fact is shown by the frequency with which he draws upon it for 
illustrations of what is requisite to a good definition. In the 
Thecetetus the definitions of square numbers, oblongs, and roots are 
used to show that enumeration is inadequate as a principle of defini- 
tion, and that definitions must be couched in general terms and must 
set off a class in accordance with a principle of logical division. 14 
In the Gorgias, rhetoric has been defined by one of the speakers as an 
art which is concerned with discourse. The looseness of this defini- 
tion is immediately noted, and it is pointed out that rhetoric has not 
been defined in such a way as to distinguish it from all the other arts ; 
for they, too, are concerned with discourse. To make the matter 
clear, an illustration is given from the sphere of mathematics. If 
arithmetic be defined as one of those arts which take effect through 
words, so also is calculation. Where, then, is the distinction? A 
difference must be pointed out — the difference being that the art of 
calculation considers not only the quantities of odd and even num- 
bers, but also their numerical relations to one another. 15 

To the sort of certainty and clearness which comes from the 
intuitive element and from careful definition in mathematics Plato 
recognizes that there is to be added that which arises from the 
method of procedure. 

Here is a science in which they distrust and shun all argument 
from probabilities. 16 " The mathematician who argued from proba- 
bilities and likelihoods in geometry would not be worth an ace." 17 

There are hints that Plato was especially interested in mathe- 
matics for its suggestiveness in respect to a particular method of 

13 Meno, 81-86. 15 Gorg., 450-51. 

14 Theat., 147-48. 16 Phado, 92. " Theat., 162-63. 



12 THE MATHEMATICAL ELEMENT IN PLATO S PHILOSOPHY 

procedure — the method of analysis. There is evidence that he paid 
special attention to this method and developed it to a high degree. 
By tradition he is credited with being its inventor. In the Meno he 
suggests that it may be applied outside of the field of mathematics. 

In arguing the question whether virtue can be taught, a hypothe- 
sis will be assumed as in geometry, and consequences deduced from 
it. If these consequences are contradictory to known facts, the 
hypothesis is rejected ; if consistent with them, it is accepted. 18 

It is not to be wondered at that a man of philosophic tempera- 
ment should have been struck with the beauty of mathematical 
procedure. At a time when fields of investigation had not been 
minutely specialized, when methods of scientific procedure were in 
the embryonic stage of development, here was a science which had 
something, at least, of a technique of its own. Starting with intui- 
tive data of undoubted clearness and with concepts unambiguously 
defined, proceeding by methods which guarded at every step against 
error, that certainty of result might be achieved which stood in 
striking contrast to the vague probabilities of other sciences. 

Closely connected with the qualities of clearness and certainty in 
mathematics are those of necessity and universality. These also are 
noticed by Plato and made a strong impression upon him. 

In speaking of arithmetic, he says that "this knowledge may 
truly be called necessary, necessitating as it does the use of the pure 
intelligence in the attainment of the pure truth." 19 This passage, 
however, is not conclusive. But in the Laivs he points out with 
reference to mathematical subjects that "there is something in them 
that is necessary and cannot be set aside ; " and he adds that " prob- 
ably he who made the proverb about God had this in mind when he 
said, ' Not even God himself can fight against necessity.' " 20 In the 
Thecetetus arithmetical notions are classed among universal notions, 21 
and in his scheme of education for the guardian class, described in 
the Republic, he makes it of great importance that attention be given 
to "that which is of universal application — a something which all 
arts and sciences and intelligences use in common — number and 
calculation — of which all arts and sciences necessarily partake." 22 

Plato's attitude with reference to the utility of mathematics is an 
interesting study. In general he deprecates the demand for utility — 
at least in so far as utility (in the practical sense) is to be taken as 

M Meno, 86-87. M Laws, 7:818. 

18 Rep., 7 : 526. 2l Theat., 185. " Rep., 7 • 522. 



PLATO S GENERAL ATTITUDE TOWARD MATHEMATICS 1 3 

the real ground of the value of the subject. He makes its value rest 
chiefly on other grounds. He takes the philosophic points of view ; 
he is always in the critical, reflective attitude of mind, or, at least, 
that attitude dominates over all others. There is abundant evidence 
of this. 

He sneers at that class of people who will consider his words as 
" idle tales because they see no sort of profit that is to be obtained from 
them." 23 The kind of knowledge in which the guardians of his ideal 
state is to be trained is not to be found in the useful arts, which 
(from the educational point of view that he has in mind) are 
reckoned mean. 2 * But they are to receive (among other things) a 
thorough training in mathematics. To this end, their arithmetic 
they are to learn " not as amateurs, nor primarily for its utility, nor 
like merchants or retail dealers, with a view to buying and selling." 
"Arithmetic, if pursued in the spirit of a philosopher, and not a 
shopkeeper," he regards as a charming science and one that is edu- 
cationally advantageous. 25 From this point of view the purer and 
more abstract the mathematics the better. He accuses the mathema- 
ticians themselves of not being altogether free from the tendency to 
look upon their science too largely from the practical side. He scores 
them for " speaking in their ordinary language as if they had in view 
practice only." They "are always speaking in a narrow and ridiculous 
manner of squaring and extending and applying and the like — they 
confuse the necessities of geometry with those of daily life ; whereas 
knowledge is the real object of the whole science." 26 

While Plato decries the insistent demand for utility and main- 
tains that there are higher values to be realized apart from the 
utilitarian standard, yet he does not fail to see the useful and signifi- 
cant place of mathematics both in the ordinary walks of life and 
also in relation to the career of the warrior. This twofold practical 
significance of mathematical study is appreciatively brought out in 
his advocacy of the teaching of children after the Egyptian fashion 
by means of mathematical games : 

This makes more intelligible to them the arrangements and movements of 
armies and expeditions ; and in the management of a household, mathematics 
makes people more useful to themselves, and more wide awake; and again 
in the measurement of things which have length and breadth and depth they 
free us from the natural ignorance of all these things which is so ludicrous 
and disgraceful. 27 

23 Rep., 7:527. 25 Rep., 7:525. 

24 Rep., 7 : 522. 26 Rep., 7 : 527. 27 Laws, 7:819. 



14 THE MATHEMATICAL ELEMENT IN PLATO S PHILOSOPHY 

Let us now take up these two points separately, beginning on 
the side of civic life. The practical importance of mathematics to 
the arts is pointed out. It is by reason of the mathematical element 
that they are enabled to rest upon a more secure basis than empiri- 
cism. Plato clearly sees that measure, the objective application of 
the principle of quantity, lies at the very foundation of all fruitful 
technical procedure. 

The arts are said to be dependent upon mathematics ; " all arts 
and sciences necessarily partake of them." 28 " If arithmetic, men- 
suration, and weighing be taken away from any art, that which 
remains will only be conjecture and the better use of the senses 
which is given by experience and practice, in addition to a certain 
power of guessing, which is commonly called art, and is perfected by 
attention and pains." 29 

In other words, he might have said that all arts are nothing but 
"cut and try" methods until application of mathematics has been 
made to them. 

In the Republic a great deal is made of the fact that mathematics 
is of practical value to the military man. 

The art of war, Plato urges, like all other arts, partakes of mathe- 
matics. 30 The principal men of the state must be persuaded to 
learn arithmetic for the sake of its military use. 31 The warrior 
should have a knowledge of this subject, if he is to have the smallest 
understanding of military tactics. 32 " He must learn the art of 
number or he will not know how to array his troops." 33 While Plato 
views knowledge as the real object of the whole science of geometry, 
as over against its practical value, yet he includes among " its indirect 
effects, which are not. small, the military advantages arising from its 
study." 34 In the scheme of education for the guardians, he says 
that " we are concerned with that part of geometry which relates to 
war ; for in pitching a camp, or taking up a position, or closing or 
extending the lines of an army, or any other military maneuvre, 
whether in actual battle or on the march, it will make all the differ- 
ence whether a general is or is not a geometrician." 35 

It is sufficiently proved that it is not from any lack of under- 
standing or appreciation of the practical value of mathematics that 
Plato decries the study of the subject for the sake of its utilitarian 

28 Rep., 7 : 522. ' Rep., 7 ■ 522. 32 Rep., 7 : 522. 3i Rep.. 7 • 5*7- 

n Philebus, 55- ai Rep.. 7-525- M Rep., 7 I 5^5. " ReP-> 7 : 5*6- 



PLATO S GENERAL ATTITUDE TOWARD MATHEMATICS 1 5 

value. He does it in order to throw the emphasis where he thinks it 
more truly belongs. He would not have the higher value ignored 
for the merely practical, which he regards as of less worth. We 
might say that with him the value of theoretical mathematics is pri- 
mary and fundamental, that of practical mathematics secondary, 
incidental, and to be taken for granted. The practical value of 
mathematics is something which he points out by the way, in 
passing; while theoretical mathematics makes an appeal to his 
deepest intellectual needs. One reason for his exalting this theoreti- 
cal study is certainly to be found in his conception of the nature of 
knowledge and of being. The discussion of that will come later; 
we are concerned here more especially with the fact — with his 
attitude toward the subject of mathematics. In this connection there 
is another important point yet to be made. 

It was in connection with the theoretical study of mathematics 
that Plato saw the possibility of scientific procedure, which was 
lacking in the empirical or merely practical. From this point of 
view we find him insisting on a sharp line of distinction between the 
scientific snd the practical, the philosophic and the popular, the pure 
and the impure in mathematics. He is interested in the pure, 
philosophic, or theoretical because it can be scientific. This funda- 
mental distinction comes out over and over again in Plato's writings. 

Knowledge, he says, is divided into educational and productive, 
the latter into pure and impure. 36 Sciences in general are divided 
into practical and purely intellectual. 37 Arithmetic in particular 
is of two kinds, one of which is popular and the other philo- 
sophical. 38 As an illustration, we may take the distinction between 
arithmetic [scientific] and calculation [popular]. Arithmetic treats 
of odd and even numbers [i. e., properties] ; calculation, not only the 
quantities of odd and even numbers, but also their numerical 
relations to one another [i. e., utilitarian values]. 39 Philosophical 
mathematics demands more careful discriminations than popular 
mathematics ; quantities which are incommensurable, for example, 
must not be confused with those which are commensurable ; their 
natures in relation to each other should be carefully distinguished. 40 
Another illustration of what is meant by the scientific study of 
mathematics as distinct from the practical or popular may be found 
in Plato's account of the properties of the number 5,040, this number 

36 Philebus, 55 ff. 38 Philebus, 56. 

37 Statesman, 258. 39 Gorgias, 451. 4n Laws, 7:819-20. 



1 6 THE MATHEMATICAL ELEMENT IN PLATO'S PHILOSOPHY 

being prescribed in the Laws as the proper number of citizens for a 
city. The number 5,040 has the property of being divisible by fifty- 
nine different integral numbers, and ten of these divisors proceed 
without interval from one to ten. 41 

Investigations into the properties of numbers yielding such strik- 
ing results as the one just cited must have profoundly impressed the 
minds of primitive thinkers. This may be at the basis of a great 
deal of the mysticism of Pythagoreanism. Plato advocates most 
strenuously the scientific study of mathematics — the study of the 
nature and properties of numbers and figures — that in mathematics 
which is exact, unchanging, absolute. Of the mathematical arts and 
sciences he maintains that those which are animated by the pure 
philosophic impulse are infinitely superior in accuracy and truth. 42 
He would make it necessary, however, for mathematical studies to be 
gone through with scientifically by a few only. 43 

In proportion as Plato admired the qualities and characteristics 
of mathematics and its possibilities in the way of achievement, 
through careful and definite methods of procedure, of certainty and 
universality; in that same proportion he also deplored the amount 
of ignorance of mathematical subjects that prevailed among the 
Greeks. Even the mathematicians themselves, he thinks, lack the 
full appreciation of the value of them when pursued in a thoroughly 
scientific manner. But he recognizes that mathematics is a difficult 
study. 

In speaking of arithmetic, he remarks that " you will not easily 
find a more difficult study and not many as difficult." 44 The difficulty 
of mathematics, the demand of mental rigor which it makes when 
pursued scientifically, may account for the ignorance of the subject 
which he characterizes as " habitual." 45 One of the chief characters 
in the Laws is represented as hearing with amazement of the Greek 
ignorance of mathematics and is "ashamed of all the Hellenes." 46 
They are so inaccurate that they are accustomed to regard all quan- 
tities as commensurable, being ignorant of incommensurables — a 
sort of knowledge not to know which is disgraceful. Also they are 
ignorant of the nature of these two classes of quantities in their rela- 
tion to one another. 47 

Three particular lines of investigation are pointed out where little 

41 Laws, 5:737-38, 745-47; 6:771; cf. 6:756. 

42 Philcbus, 57. " Rep., 7 : 526. 49 Laws, 7:819. 

43 Laws, 7: 817, end. K Laws, 7: 818. "Laws, 7: 819-20. 



PLATO S GENERAL ATTITUDE TOWARD MATHEMATICS 1 7 

really scientific work in mathematics has yet been done. Little seems 
to be known about solid geometry ; no director can be found for it, 
and none of its votaries can tell its use. The subject is declared to 
be in a " ludicrous state." 48 Secondly, the mathematical study of the 
heavens is a work, he says, infinitely beyond our present astrono- 
mers ; 49 and thirdly, in the study of harmony, even by the Pythago- 
reans, the procedure is not mathematical enough, for problems are 
not attained to. 50 All of these subjects, Plato feels, are as yet too 
empirical. 

The philosophic point of view here as well as elsewhere dominated 
his attitude. He lent the whole weight of his influence to the develop- 
ment of these subjects along theoretical, scientific lines. 51 

Judged by the standard of original solutions, doubtless it is a 
correct estimate of Plato to say that he was not a mathematician, 
yet he has a positive contribution to make, and that too of a character 
which ought to rank with the extension of the science by means of 
original solutions. This contribution was made through the reaction 
of his philosophic insight upon the technique of mathematics. The 
critical faculty of the philosopher was very much needed just at that 
time in this field of investigation. We must remember that both 
arithmetic and geometry were in a very fragmentary condition. It 
was before the time of Euclid's Elements. Mathematics could not 
with propriety be said to be organized. It was still decidedly crude. 
Some difficult and very complex problems had been solved, to be 
sure. This is rather a basis for admiration of Greek genius and the 
intellectual power of some few individual mathematicians than for 
inference as to the high development of mathematical science. 
Mathematics, and Plato felt this, though the most exact and con- 
sistent of any body of knowledge, yet was scarcely worthy of the 
name of science, so much was it a body of empirical results and 
disjecta membra. 

The progress of mathematics does not consist alone in lines of 
investigation which lead to new solutions of problems. These them- 
selves depend upon modes of procedure. These modes of procedure 
are at first not differentiated from the solutions in which they occur, 
they are not generalized. Each problem has, as it were, an inde- 
pendent character — its solution is particular and peculiar to itself. 
Reflection upon the process reveals general principles and leads to 

^Rep., 7 : 528. 50 Rep., 7 : 530-31. 

49 Rep., 7 : 530. S1 See note at end of this chapter. 



1 8 THE MATHEMATICAL ELEMENT IN PLATO'S PHILOSOPHY 

the formulation of method. This is the work essentially of the 
philosophic mind. It is just here that the greatest significance of 
Plato to mathematical science comes in. To a genuine interest in 
and familiarity with mathematics he added the philosophic interest 
just at the time when mathematics had progressed to that stage of 
development in which the next step necessary to further progress was 
the analysis of its concepts and processes and the formulation of its 
technique. This technique, when developed, could be directed back 
by the specialist upon the great unsolved problems with an added 
power which enabled him to secure further and more striking results 
in his field of original mathematical investigation. In this way the 
philosopher equally with the mathematician becomes a contributor 
to the advance of mathematical science, and it is difficult to determine 
which of the two is more truly the mathematician. Certainly it was 
Plato's philosophic temper of mind that made him "the maker of 
mathematicians." 

The '" Eudemian Summary " states that M Pythagoras changed 
the study of geometry into the form of a liberal education ; for he 
examined its principles to the bottom, and investigated its theorems 
in an immaterial and intellectual manner." 52 Even if we can rely 
upon this statement as authoritative, still it is true that there remained 
a great work to do in the way of putting mathematics upon a 
thoroughly scientific basis. Even with the Pythagoreans there 
remained much of the mystical element, which drew attention away 
from the natural fields of mathematical investigation and was a 
hindrance to legitimate scientific development. Outside of Pythago- 
reanism rational and empirical results were apt to be very loosely 
discriminated, and to the empirical result there was attached a blind 
and unjustifiable worth. This may be illustrated by the old Egyp- 
tian method of finding the area of an isosceles triangle, which, among 
other rules drawn from the Ahmes papyrus, passed current in 
Greece. According to this rule, the area of the isosceles triangle was 
found by taking one-half the product of the base and one of the 
equal sides. Of course this would be an exaggeration of the con- 
dition of affairs as it existed in the time of Plato. But we may 
judge from the scoring which he gives empirical methods that 
instances of procedure of this sort were still frequent enough. 

Now, Plato was especially enthusiastic over the scientific possi- 
bilities ot mathematics. From this point of view definition was of 

52 Gow, p. 150. 



plato's general attitude toward mathematics 19 

great importance. Plato had learned from Socrates the importance 
of analyzing and denning concepts in ethics. He applied the prin- 
ciple to mathematical science, insisting upon the most careful investi- 
gation of its fundamental concepts, resulting in a more rigid and 
precise formulation of its definitions and axioms. Whether Plato 
actually completed any considerable amount of this work or not, there 
can be little doubt that his influence in the matter was a decisive 
factor in that reconstruction of geometry which soon culminated in 
Euclid's Elements — a formulation so exact and comprehensive that 
for many centuries it remained the text-book of the civilized world 
and is able still to infuse its spirit into every modern school text in 
geometry. 

To this result Plato contributed largely in another important 
respect. Noting the possibilities of exactness, rigidity, and necessary 
conclusions in mathematical procedure and reflecting upon and uni- 
versalizing its processes, " he turned the instinctive logic of the 
early geometers into a method to be used consciously and without 
misgiving." 53 It is worthy of note that Plato seems to be equally, 
if not more, interested in methods than in results. One cannot read 
carefully the demonstration with the slave boy in the Meno 54 without 
noticing this fact. Plato is intensely interested in the reasoning 
process. This point will be emphasized again in another relation 
(see p. 47). Moreover, in this passage in the Meno he points out the 
mathematicians' use of hypothesis, which is none other than the 
method of analysis. It will later be shown (see p. 57) how fully 
conscious of the essential elements of this method Plato became. 
Whether the invention of the method be attributed to Plato or not, 
there is little doubt that the tradition which ascribes it to him rests 
upon the fact that he successfully developed and used the method as 
a powerful instrument of investigation. 

The tendency of all this improvement in the direction of rigor of 
definition, careful sifting and clear statement of postulates, analysis 
and generalization of process, and formulation of logical methods, 
was to give mathematics a technique and make it more scientific. In 
this same direction tended the determination and limitation of fields 
of investigation. Problems in geometry were limited to those capable 
of construction by ruler and compass. The study of solid geometry 
was encouraged. 55 Astronomy and also harmony were to be made 
mathematical in character. 56 The reaction of philosophy upon 

53 Gow, p. 175. 5i Meno, 81-86, 86-87. 55 Rep., 7 : 528. 58 Rep., 7 : 530-31. 



20 THE MATHEMATICAL ELEMENT IN PLATO S PHILOSOPHY 

mathematics in Plato was certainly an important factor in making 
this subject scientific in character. In Plato you find no patience with 
empirical methods and empirical results. Nor do you find in him any 
but the slightest traces of a tendency to make a mystical use of 
mathematics. 57 His demands are over and over again for the 
theoretical, pure, and scientific as against the practical, popular, and 
empirical. He scores the mathematicians themselves for not being 
scientific enough, the students of astronomy and harmony for not 
being mathematical enough. Whether he was a mathematician him- 
self in the ordinary sense of the word, or not, he certainly made a 
contribution to the subject of mathematics from the philosophical 
point of view and set the ideal which mathematicians had henceforth 
to follow in the pursuit of their science. 

The writer originally worked out in considerable detail the question of 
Plato's relation to the mathematicians of his time and the extent and char- 
acter of his influence upon the progress of mathematics. But this ground has 
been so thoroughly covered by the great historians of mathematics that he 
has thought best to give only a brief general statement of the significance of 
Plato to mathematics from the philosophical point of view. For further 
details as to mathematics proper the reader is referred to the bibliography at 
the end of the book. 

67 For these instances see Rep., 8 : 546 ; Timaus, 35-36, 38 ff., 43, 53 ff. These 
may be less mystical than they appear to be. See p. 40 of this book. 



CHAPTER II. 

THE FORMULATION OF PHILOSOPHICAL PROBLEMS. 

The philosophy of Plato grows out of a highly complex situation 
involving many mutually interacting factors both personal and 
environmental. In analyzing out a few of the most significant and 
determining strands of his thought, it is not necessary to assume that 
all of them thus analyzed out were consciously determining in the 
mind of Plato himself. Quite commonly the most fundamental fac- 
tors in a man's thought are so much a part of his whole attitude and 
integral mode of reaction that he is entirely unaware of them as 
determining in his mental processes. Yet another, viewing them 
from the outside, may clearly see, interpret, and point out their 
psychological and logical bearing. Whatever attitude the reader may 
take with reference to the point of view running through this book, 
the character of the work as an attempt to analyze after the fact 
must not be overlooked. Isolation of parts for the sake of getting 
their bearing and seeing their significance gives both organization 
and emphasis which did not belong necessarily to the work of Plato 
as he conceived it himself. 

The problems of no great thinker arise in his consciousness ab 
externo and ex abrupt o; they have some connection with his imme- 
diate environment, social and intellectual. A period of reaction to or 
against the philosophic ideas of others naturally precedes definite and 
conscious formulation of one's own. Such reaction is both the 
stimulus to initiation and the condition for progress. The discussions 
of the dialogues show that Plato familiarized himself with all the 
leading historical and contemporary philosophical systems that found 
currency in Greece. The details of these systems are not up for our 
consideration here ; a certain familiarity with them will have to be 
assumed. We can touch only upon certain characteristic concepts as 
they affect the understanding of our special problem. 

The speculations of the earlier philosophers had resulted in fixing 
attention upon certain great limiting concepts. Especially did the 
great opposing attitudes of the Eleatics and the Heracliteans domi- 
nate thought in such a masterful fashion that no serious and far- 
reaching reflection was possible without taking into account the 



22 THE MATHEMATICAL ELEMENT IN PLATO S PHILOSOPHY 

problems involved in the antitheses of being and becoming, of the one 
and the many, of permanence and change, of essence and genesis, of 
sensation and thought, of opinion and knowledge, of appearance and 
truth. At first interest centered largely in the external and objective 
world, the problems were those of cosmology and ontology. The 
problems of man — questions regarding the soul, the mental pro- 
cesses, human activities and conduct — were incidental. When dis- 
cussed, the tendency was to treat them from the point of view of man 
as a part of the cosmos. They were taken up from the same objective 
point of view which dominated the nature-philosophy. The raison 
d'etre of interest in these problems seems to have been very largely 
that without reference to them the cosmological account would have 
been incomplete. This was as true of atomism and other mediating 
systems as of Eleaticism and Heracliteanism. 

The profound social and political disturbance incident to the 
Persian wars disrupted the routine of the old Greek life and shifted 
the center of attention and of interest from cosmology to human life. 
The significance of man was brought to consciousness — his achieve- 
ments, his powers. The growing importance and scope of the 
political activity carried in its train a great stimulus to the study of 
rhetoric and eloquence. Problems of human mind and of human 
conduct were brought to the focus of attention. Quite naturally, with 
the rise of a new set of problems, the intellectual tools forged in 
dealing with the old questions were tried upon the new ones. Points 
of view, fundamental distinctions, working concepts characteristic 
of the departing age, were drawn upon in the attempt to define and 
solve the problems of the new era. 

So far as the particular Platonic problem of this book is con- 
cerned, the first movement along the new line to demand our attention 
is that which has come to be quite ambiguously associated with the 
name " Sophists." What I have in mind is the philosophy of rela- 
tivity, by whatever name called, or with whatever individual asso- 
ciated in thought — the " flowing philosophy," as Professor Shorey x 
has quite aptly styled it — an outgrowth of the Heraclitean doctrine 
of " flux " and the sensationalistic psychology of Protagoras. I shall 
hereafter refer to this type of philosophy as Protagoreanism. 

Protagoras applied the Heraclitean principle of motion to the 
analysis and explanation of perception. The result was a thorough- 
going doctrine of the subjectivity and relativity of sense-perception. 

1 Unity of Plato's Thought. 



THE FORMULATION OF PHILOSOPHICAL PROBLEMS 23 

Man has already been identified with a particular phase of the cosmos. 
The principle that had explained the universe Protagoras extends 
more fully than his predecessors to the explanation of man. World- 
process and mental process are identified throughout the whole of 
man's mental life. Sensation and thought, opinion and knowledge, 
are continuous phases of one world-process, the resultants of the 
interactions of continually shifting motions. Knowledge is percep- 
tion ; the relativity of perception is the relativity of knowledge. In 
bringing to consciousness the principle of subjectivity and in viewing 
the psychical life from the side of process, Protagoras made a very 
significant contribution to psychology ; but he failed to find within 
the process any solid basis for the validity of thought. When the 
Protagoreans applied the principles of this " flowing " philosophy to 
the concepts of ethics, the fixity and permanence of the solid structure 
of habit, custom, and tradition, in which the morality of the age 
inhered, was reduced to a fleeting, fluctuating stream of mere con- 
vention. Ethics, like knowledge, was subjective and relative. The 
world of things, the world of experience, the world of conduct, were 
all alike subject to the Heraclitean law of "flux," genesis, or 
becoming. 

Socrates was not deeply interested in the speculative problems of 
physics or of ontology. There is no reason to think that he reacted 
especially against the Protagorean philosophhy. But he did react 
against the situation of ethical confusion and moral relaxation of his 
day, which found aid and comfort in the negative and relativistic 
type of philosophy. His moral earnestness could not endure the 
destruction of the ethical concepts. These must be restored. If the 
moral sanctions inherent in faith in the old regime had been loosed 
from their moorings, then they must be grounded anew. Socrates 
sought to give the virtues a securer basis than convention or habit 
by grounding them in knowledge. Not everything was under the 
law of change ; there were such things as universals. This he sought 
to show not upon the basis of any theory or speculation, but upon 
the basis of an examination and analysis of the facts of human con- 
duct. He found that the artisan, at least, had a standard of the 
good. The shoemaker, the harness-maker, the shipbuilder, the maker 
of weapons, etc. — in fact, every artisan — worked toward seme 
standard of excellence, even though he may not have set that standard 
for himself with reference to a more ultimate end. The success of 
these men in attaining the good within their limited and circum- 



24 THE MATHEMATICAL ELEMENT IN PLATO S PHILOSOPHY 

scribed sphere depended upon their having knowledge; this one 
thing they knew. With them their knowledge and their virtue, or 
excellence, were one. The great trouble with the politician, or states- 
man, was that he did not know what was the good of the state, for he 
did not know the nature and the end of the state. The great trouble 
with people in general was that they acted upon the basis of con- 
vention or habit, unconscious of the principle in accordance with 
which they were acting, thinking themselves wise when they were 
really ignorant. So Socrates conceived it to be his mission to ques- 
tion people till he could show them their ignorance and make them 
seek to become wise. The great ethical significance of Socrates lies 
in the fact that he made morality a personal thing, not a conventional 
thing. Knowledge of ends, not imitation, or tradition, or custom, 
was its basis. He recognized the subjective factor, but not in the 
same way as the Protagoreans. 

As with Socrates, so with his pupil Plato, his dominant interest 
was ethical and practical. This point of view I would maintain in 
spite of the fact that Plato devotes much time and space to the 
discussion of many abstract and abstruse metaphysical questions. He 
had, undoubtedly, a fondness for theoretical questions ; but, as a 
rule, their discussion is for the purpose of throwing greater light 
upon some ethical or other practical human problem. Plato took up 
the ethical point of view of Socrates which made virtue a function 
of knowledge. But he pushed the analogy of the arts much farther. 
Nor was he content to let the theoretical question raised by the Pro- 
tagoreans go untouched. If the virtues rest on a basis of knowledge, 
as Socrates contends ; and if at the same time knowledge is sense- 
perception and a relative thing, as the Protagoreans contend, then 
Socrates is in as sorry a plight in the matter of finding secure ethical 
sanctions as when he began. The basis of ethics is insecure as long 
as it abides within the sphere of becoming. The ethical demand, on 
logical a priori grounds, is for knowledge which is of the eternal 
and abiding. The question for Plato is, then : Is there any such 
knowledge ? The solution of the ethical problem leads him over into 
the epistemological question. 

Protagoras, under the impulse of the Heraclitean factor, has 
identified sense-perception and knowledge. Plato, in order to give 
ethics a secure logical foundation, will again recognize the Eleatic 
factor and set up a distinction between sense-perception and knowl- 
edge, bringing both factors within his own system, with a decided 



THE FORMULATION OF PHILOSOPHICAL PROBLEMS 2$ 

emphasis on the value for knowledge of the Eleatic factor. He 
admits in general the inadequacy of sense-perception and seeks to find 
elsewhere a more secure basis for knowledge. But as for the total 
relativity of sensation, there is at least an intimation in the Thecetetus 
(171) that he does not think that the doctrine is true. In the 
Republic (7:523) he has this positive statement: "I mean to say 
that objects of sense are of two kinds ; some of them do not invite 
thought because the sense is an adequate judge of them ; while in the 
case of other objects, sense is so untrustworthy that further inquiry 
is imperatively demanded." The nature of this further inquiry will be 
taken up later. What I want to bring out here especially is the 
fact that Flato does give the senses some positive function, but at the 
same time he would not make sense-perception the equivalent of the 
whole knowledge-process. There is the question of the adequacy and 
the inadequacy which must be settled by some higher function. A 
distinction has to be set up between the lower and the higher, between 
sensation and thought. 

Furthermore, Plato contended that there is knowledge which does 
not come through the senses. This point he works out in the 
Thecetetus. The senses are specific — the eye being concerned with 
seeing, the ear with hearing, etc. But the common notions which we 
have are not thus specific in character. Our knowledge contains ideas 
of being, or essence, and of non-being, of likeness and unlikeness, of 
sameness and difference. Ideas such as these — abstract, universal, 
or embodying the results of comparison — cannot have come through 
any bodily sense ; they have been perceived by the soul. 2 Thus there 
must be a distinction made between the senses and the intellect. 
Knowledge is not necessarily identical with sense-perception ; it may 
have its basis in a higher faculty and have a character of permanence 
and stability characteristic of the world of being as opposed to that 
world of becoming which finds conscious expression through the 
process of sensation. 

We are now at the point where we can begin to study specifically 
the significance of the mathematical element in Plato's thought. On 
logical grounds, the ethical demand is for a source of knowledge not 
subject to the Heraclitean law of "flux," or becoming. The Pro- 
tagorean position of the identity of sensation and knowledge must 
then be overthrown. This is done by setting up a distinction between 
sensation and knowledge. It has been argued that there is a kind of 

2 Thea>t., 184-86. 



2b THE MATHEMATICAL ELEMENT IN PLATO'S PHILOSOPHY 

knowledge which is not of sensational origin, and also that w T here 
sensation is involved the basis of knowledge may lie in the exercise of 
a higher faculty. Both of these points Plato followed up by an 
appeal to mathematics. It may even be that it was mathematics 
which gave him his first clue to this line of argumentation. Cer- 
tainly the bringing in of the argument from mathematics made the 
justification of his position so clear and striking that it had all the 
force of a new proof rather than one of the same nature. 

The best place to make a beginning of the mathematical argu- 
ment will be a passage in the Republic. This may be summarized as 
follows : Objects of sense are of two kinds : ( i) those of which sense 
is an adequate judge, and which hence do not invite thought; (2) 
those of which sense is not an adequate judge, and which hence do 
invite thought. The second case is that of receiving opposed impres- 
sions at the same time from the same object; e. g., to the sense of 
touch hard and soft at the same time ; or to the sense of sight great 
and small. Thus a conflict is created. This sense-conflict marks the 
beginning of an intellectual conflict. Since two qualitatively distinct 
and opposed impressions have been received, the problem arises as to 
whether they can come from one and the same object, or whether 
there are not two objects. The soul is put to extremity and summons 
to her aid calculation [an intellectual principle] to determine whether 
the objects announced are one or two ; and hence arises the distinc- 
tion between the perceptible and the intelligible. When mind has 
come in to light up, analyze, and interpret the conflicting manifold, 
of which sense is not an adequate judge, the conception of the one 
and the many both arise, and thought is aroused to seek for unity. 3 

According to Plato, then, the distinction between the senses and 
the intellect arises through a process of reflection stimulated by a 
sense-perception situation involving contradictory and conflicting 
experiences. This situation can be resolved only by the introduction 
of the mathematical process. But when this process is once intro- 
duced the distinction between sense and intellect is already under 
way. The mathematical thinking does not begin so long as there is 
only a confusion of sense-experience, but only when an intellectual 
conflict h?s been provoked and the mind has been put into the inquir- 
ing attitude. Furthermore, these mathematical notions, though 
brought to light under the stimulus of a certain type of sense- 
experience, are not themselves of sense-origin. They could not be; 

: Rep., 7 : 523-25- 



THE FORMULATION OF PHILOSOPHICAL PROBLEMS 2*] 

for the senses are specific. Plato finds no separate sense organ for 
them, and his conclusion is that they are perceived by the soul alone. 4 

Thus mathematical thinking originates and necessitates the dis- 
tinction between the senses and the intellect; for no mathematical 
thought would be possible without such distinction. But it would 
never occur to the mind of Plato to doubt that we do have a genuine 
knowledge-process in mathematical thinking. The logical a priori 
demand for the overthrowing of the Protagorean position by a 
reassertion of the distinction between sense-experience and rational 
process receives specific content when Plato turns to the study of 
mathematical thought and observes what takes place there. It is 
found to be justifiable and necessary from the point of view of an 
accepted and undoubted realm of knowledge. Yet we are not war- 
ranted on the basis of this passage from the Republic in saying that 
Plato conceived of the distinction as an absolute one in the broadest 
sense of the word " knowledge." 

Another passage in the Republic is clearer still in showing the 
distinction between the sensible and the intelligible as the effect of 
the mathematical element. Also it throws some light upon the 
working nature of the distinction. This passage may be summarized 
as follows : 

The body which is large when seen near appears small when seen 
at a distance. And the same objects appear straight when looked at 
out of the water and crooked when in the water; and the concave 
becomes convex, owing to the illusion about colors to which the 
sight is liable. Thus every sort of confusion is revealed within 
us. But the arts of measuring and numbering and weighing come 
to the rescue of the human understanding, and the apparent greater 
or less, or more or heavier, no longer have the mastery over us, but 
give way before calculation and measuring and weight. And this 
surely must be the work of the calculating and rational principle in 
the soul. And when this principle measures and certifies that some 
things are equal, or that some are greater or less than others, then 
occurs an apparent contradiction. But such a contradiction is in 
reality impossible — the same faculty cannot have contrary opinions 
at the same time about the same thing. Then that part of the soul 
which has an opinion contrary to measure is not the same with that 
which has an opinion in accordance with measure. The better part 
of the soul [i e., intellect, or reason] is likely to be that which trusts 

i Thecetetus, 185. 



28 THE MATHEMATICAL ELEMENT IN PLATO'S PHILOSOPHY 

to measure and calculation. And that which is opposed to them 
[/. e., sense-perception j is one of the inferior principles of the soul. 5 

Here it is shown that the mathematical principle of measure in its 
various forms brings in the element of intellectual control, and that 
where this control is introduced we have greater certainty than can 
be derived merely from the senses. This ordering and controlling 
function of mathematics will receive further discussion later in con- 
nection with the analogy of the arts. I point it out here merely to 
suggest that this gives us an indication that Plato works out the 
distinction between the senses and the intellect, not merely for the 
sake of maintaining a rigid separation between the sensible and the 
intelligible, but that he may find a higher principle by which to judge 
and control the lower. The cognitive aspect of that which takes 
mathematical form is very different from the cognitive aspect of that 
which takes merely perceptual form. We can see that in the mind 
of Plato not only does mathematics effect the distinction between the 
sensible and the intelligible, but he also intimates that the presence of 
the mathematical element is criterion of the value of a thing as 
knowledge. When Plato once gets this view of mathematics, it 
transforms his whole conception of the arts and sciences, as we shall 
see later. It also has very significant ontological implications ; for in 
Plato epistemology and ontology are very closely bound up together. 
I may point out in passing that the view of the mathematical element 
as criterion of value for knowledge serves as a basis for the doctrine 
of idealism. 

So closely interwoven are the strands of Plato's thought that 
from this point on we might follow them up in any one of several 
different ways and our problem work out very much the same. How- 
ever, as the analogy of the arts plays such a fundamental part in all 
his thinking, it may be well to work that out in part at this point It 
was in connection with the problem of ethics that the analogy of the 
arts made such a profound impression upon the mind of Socrates. 
This was also the most vital spring of Plato's interest in the arts and 
the artisan class. The point which is significant for us at the present 
is that both Socrates and Plato saw definitely in the arts the realiza- 
tion of the good dependent upon some measure of knowledge — 
knowledge at the very least of immediate ends. Socrates stated the 
principle, but we cannot tell how far he worked out its rationale and 
technique. Probably not very far. Plato pressed the analogy of 

p., io : 602-3. 



THE FORMULATION OF PHILOSOPHICAL PROBLEMS 29 

the arts to take in more and more remote ends, and he also worked 
out the means side of the problem of the arts in both its practical and 
its epistemological aspects. It will be easier, because more natural, 
to get at the logical significance to Plato of the analogy of the arts 
by beginning with the practical aspect. Following up the ontological 
problem for a little while will throw light upon the logical, or 
epistemological, one. 

On the ontological side, what is common to all the arts is the 
fact that they are concerned with production. This is true not alone 
of the simple industries, but also of the ruler of the state. His art, 
too, is concerned with production. 6 Now, production involves 
motion, the destruction of that which exists in one form by the 
breaking of it down or dividing it up and making new aggregations 
or some change in the relation of parts so as to produce a change of 
form. It is a process of becoming. The arts seem to fall wholly 
under the Heraclitean law of "flux," yet here Plato will find an 
Eleatic element of the abiding. On the lowest level that which is 
produced may come to be what it is by some chance, or by the happy 
guess of somebody, but this is not art. 7 Art involves the exercise of 
some principle of control. The " cut and try " process is not art, nor 
is mere routine art. Production as an art is not a random matter, 
but is in acordance with mathematical principles. 8 All arts and 
sciences necessarily partake of mathematics. 9 Mathematics intro- 
duces the element of intellectual control into the process of produc- 
tion. The flowing sense-world is subjected to measure in all its 
various forms, and thus made subject to a higher world of order, 
beauty, and harmony. We do not have merely a world of becoming 
in all its ungraspableness, nor a world of unitary pure being in all its 
lonely grandeur. In the arts the two limits are brought together 
through the mathematical element into one ordered whole. 

Now, we want to get the intellectual significance of this. The 
arts, all the processes of production, are concerned immediately or 
remotely with the satisfaction of human wants. The word " want " 
is ambiguous, and in its very ambiguity it is true to the situation to 
which it applies. There is both a physical and a psychical implica- 
tion. On the lowest level the satisfaction of a want involves a need 
and the meeting of that need all within the unity of the same act 
without any process of intermediation between the two limits. But 
when the want is not satisfied by an immediate response to stim- 

6 Statesman, 261. 7 Fhilebus, 55. 8 Statesman, 284. 9 Rep., 7 : 522. 



30 THE MATHEMATICAL ELEMENT IN PLATO'S PHILOSOPHY 

uli, then the need takes the psychological form of consciousness 
of a lack. In ontological terms this would correspond to not-being. 
The tension of this situation may be relieved by some chance or ran- 
dom activity or by some process of external imitation. This repeated 
gives habit or routine. The process of production under these cir- 
cumstances would be wholly empirical. There is a certain sense in 
which we would then have arts. There is a certain way which 
the workman has of reaching an end ; there is a certain sense in 
which he may be said to know how to get a certain result. Yet 
Plato would not call this art. The process is not intellectually con- 
trolled. It is mathematics which introduces this control. Response 
then does not follow immediately upon stimulus, nor does it flow off 
without attention into some routine channel. On the psychological 
side as well as on the physical, the process of production is mediated 
and controlled with reference to an end. The states of consciousness 
are not a mere "flux.'" They become ordered, arranged, in accord- 
ance with a principle. We have technique instead of routine ; con- 
trol, or power, instead of chance ; rational method instead of habit, 
custom, and imitation. On the psychical side as well as on the 
physical, the processes of production are no longer mere becoming, 
but arts (T*x vr i)- This involves not only knowledge of an end, 
but also that form of intellectual control which takes up means and 
end and consciously identifies them within one process through the 
intermediation of a regular series of steps. This is what mathematics 
enables one to do with the process of production. It gives knowledge 
and control of process with reference to ends. 

Whether or not Plato was able to work out the complete psychol- 
ogy of the technical arts, he certainly did get a great deal of their 
intellectual significance. Through a comprehension of the meaning 
of the mathematical element the analogy of the arts became less of a 
mere analogy to him than it had been to Socrates. He had in his 
mind a clear working image of a type of intellectual control — an 
image rich in suggestion as to the possibility of a knowledge higher 
than sense, which could hold in its grasp and unify the fleeting and 
fluctuating sense manifold. 

On the basis of the psychology of the industrial arts, involving 
the intellectual principle of mathematical control, Plato would arrive 
again at the conception of a distinction between the senses and the 
intellect — this time in a clearer and more concrete form than that 
already pointed out in the illustration of the mathematical element 



THE FORMULATION OF PHILOSOPHICAL PROBLEMS 3 1 

coming in to settle the conflict of sense (see p. 26). The world of 
experience, on its cognitive side, would fall into two main divisions 10 
— all that which is matter of opinion (8d£a) and all that which is a 
matter of rational process or intellect (vo'^oW). On the objective 
side this distinction would correspond to that which is the object of 
sense-perception (to bparov — the visible used here as a symbol for 
all the perceptible) , as over against that which is the object of thought. 
(to votjtov). Let us see how this worked out from a study of the 
arts. In the first place, those who worked by routine or rule of 
thumb could give no reason for their method of procedure; they 
could not see it in the light of any rational principle. From this 
point of view, they were entitled only to an opinion. They were 
either following the rule of another, or being guided by a series of 
associations of sense-experiences through which they had passed 
before in getting the same result. Even though they might be 
engaged m a real art which had a technique, which had already been 
brought under the law of intellectual control through the introduction 
of the mathematical element, yet that technique might be, so far as 
their own consciousness was concerned, mere routine, and they 
might not themselves be conscious of any rational principle of con- 
trol. If so, they could not be said to have knowledge in the higher 
sense, but only opinion. This was the case with the vast majority of 
the artisan class, and it was this fact that made Plato rank them so 
low as he did. Their art, to be sure, was based on principles of the 
higher knowledge, but they themselves had not this knowledge. It 
could be seen that in the arts knowledge involved not merely the 
ability to reach a certain end, but also insight into the process by 
which that end could be reached, the ability ideally to construct that 
process with direct reference to the end and to intellectually control 
it. Opinion, even at its best, is not knowledge. The man may have 
true opinion in the matter of his process of production, yet there is a 
distinction between the cognitive aspect of his consciousness and that 
of the man who knows the rationale and can construct for himself the 
method. The former is a perceptual type, the latter a noetic type, 
of consciousness. 

We have gone far enough now to see how Plato came to the con- 
ception of a type of knowledge higher than sense-perception. We 
have also seen how large a part mathematics played in his work of 
transcending the Protagorean-Heraclitean epistemology. This work 

10 Here I use the terminology of Republic, 6: 508-11. 



2,2 THE MATHEMATICAL ELEMENT IN PLATO'S PHILOSOPHY 

is, indeed, not yet complete ; but, as the ethical and the logical prob- 
lems are so closely intertwined in the thought of Plato, it may be well 
to gather up some of the ethical implications at this point. In fact, 
by so doing we shall the better see what was the impetus which 
drove Plato to carry out his epistemology to the limit of dialectic. 

According to the Socratic formula, moral conduct, the attain- 
ment of the good, is a function of knowledge. This he illustrated 
by the analogy of the arts. Plato has taken up this analogy and 
done two things : ( I ) he has shown that the knowledge on which the 
arts rest is of a higher type than that of sense-perception, involving, 
as it does through the mathematical element, the power to judge and 
control the sense manifold ; (2) he has analyzed the process of 
attainment of the good in production, and has found that the signifi- 
cance of the cognitive factor involved consists in the fact that here 
is used a rational method, or technique, made possible by mathemat- 
ics, for the intelligent adaptation of means to an end. Applying the 
results of this analysis to the problem of ethics, it is not enough to 
say that virtue is knowledge, not even if you say knowledge of ends. 
Scientific ethics must meet a further demand. Conduct, if it is to be 
regarded as ethical in the scientific sense, must involve that higher 
type of knowledge which is conscious of its own technique, and can 
hence control the elements of a situation in such a way as to be sure 
of producing the good, and not merely guess at it, or run the risk of 
failure through the breaking down of habit or routine. 

Socrates and Plato both observed all around them the good 
existing in isolation from any principle of propulsive power, not 
brought under the control which comes from knowledge. Charmides 
was temperate, but he did not know what temperance was ; Lysis 
was a friend, but he could not define friendship ; etc. Thucydides 
and Aristides were noble in their deeds, but they did not know how 
to impart nobility to their sons, Melesias and Lysimachus. 11 Why 
could they not teach it? Socrates maintained that virtue is knowl- 
edge ; and, if knowledge, then it can be taught. Plato showed the 
conditions which must be satisfied in order for virtue to be taught. 
It must be a virtue that is not wholly imbedded in habit, routine, or 
custom. Its rationale must be known, the technique of its process 
must be worked out. Education is a process of production ; the 
teaching of virtue, like the teaching of an art, implies the ability to 
control a process, and control of the process, in any scientific sense of 

11 Lac'ics, 178-79. 



THE FORMULATION OF PHILOSOPHICAL PROBLEMS 33 

the word, implies knowledge of its technique. The practice of virtue 
involves the same principle. The man whose moral conduct is 
regulated merely by convention is on the same level of opinion and 
empiricism as the artisan who depends wholly upon routine. Each, 
in his own sphere, may attain the good ; but that result is uncertain, 
insecure, liable to all sorts of error. Only the man who can control 
the process has virtue in any true or scientific sense of the word. 1 
may have gone beyond the words of Plato, but I have tried to inter- 
pret his problem in the spirit of his words. 

There is still one other knowledge-factor in ethical conduct 
besides knowledge of the technique. That is knowledge of the end. 
Socrates brought this out, and Plato emphasized it. The artisan, 
even when he rises to the higher plane of having a rational under- 
standing of the technique upon which his art rests, still may be on 
the lower ethical plane. He is limited in respect to his knowledge 
of ends. What he makes he may make with intelligent adaptation of 
means to ends, so that, with reference to the end that he has in view, 
it may be perfectly good. But whether it is good in any further, 
more remote, or ultimate sense he does not know. What he makes 
he turns over to another to use. He may make the shoe, but 
whether it is good to wear a shoe is outside of his province. The 
physician may by his art know how to save the life, but he does not 
know whether it was better for the man to have lived or to have 
died. The pilot may carry you safely across the water, but it may 
have been better that you should have drowned. Instances of this 
sort Plato multiplies almost without number. A completely scientific 
ethics must take into account both knowledge of ends and knowl- 
edge of process. From this point of view, we can understand 
Plato's numerous thrusts at the sophists, the lawyers, and the poli- 
ticians. The sophist professes to teach rhetoric, eloquence, virtue; 
but when examined he knows neither the nature, the true end, nor 
the technique of these. The politician would make laws for the state ; 
but he does not know what justice is nor the process of attaining it. 
The ambitious man would rule, but he knows less about the nature of 
the kingly art than the cobbler does about making shoes. We call in 
specialists to judge of a musical instrument, a piece of armor, a case 
of sickness ; but we are asked to turn over the larger interests of 
education in morals and the conduct of the state to men who know 
neither the process nor the end of the art which they are willing to 
undertake. Plato's ethical demand is that virtue shall rest upon 



34 THE MATHEMATICAL ELEMENT IN PLATO S PHILOSOPHY 

knowledge through the whole length and breadth of human activi- 
ties. The system of human relations, social, industrial, and political, 
should fall into an order in which the highest class should be an 
embodiment of the ideal of the completely scientific standard of 
ethical conduct. Illustrations of some of the above points may be 
found in Gorgias, 455 and 511. 

Complete and reciprocal knowledge of ends and technique is not 
found in the case of any of the arts, neither has it been revealed in 
the mathematical element which lies at their basis as a principle of 
intellectual control. The mathematical element has revealed only 
the possibility of intellectually controlling the process of realizing an 
end when it is already known. It cannot tell us whether or not the 
end is good in light of a further principle. The epistemological 
problem has not yet been pushed far enough for Plato fully to 
establish and ground ethics upon the rational and scientific basis 
which he demands. We have come to the end of the ethical problem 
until we can push farther the logical problem by taking up the 
method, or technique, of knowledge in general. 

Before passing on to the method, it may be well to go back and 
work out some of the further implications, already hinted at, of 
Plato's conception of the significance of the mathematical element. 
There are three principal topics which will come up for considera- 
tion : the influence of mathematics upon his conception of the 
sciences, the place of mathematics in his cosmology, and the relation 
of mathematics to idealism. 

The sciences in Plato's day were in their infancy, the organiza- 
tion of knowledge could scarcely be called scientific anywhere except 
in some parts of mathematics, and even in mathematics there was 
much that was wholly empirical. Yet from mathematics Plato got the 
conception of what intellectual control of material meant. We have 
already pointed out that (see p. 28) this intellectual control involved 
an Eleatic factor in knowledge, which made it superior to the law 
of the " flux " of the senses. Through the mathematical element 
something abiding and valid and universal was attained. The 
organization of other departments of knowledge, Plato conceived, 
could be made scientific, if procedure was based on mathematical 
principles, if measure and number were introduced. It was from 
this point of view that he criticised the study of harmony and of 
astronomy as it was conducted in his day. Astronomy must be 
something more than Star-gazing in order to be scientific. The 



THE FORMULATION OF PHILOSOPHICAL PROBLEMS 35 

heavenly bodies are conceived as themselves moving according to 
mathematical laws. The proper method of arriving at astronomical 
truth is by attacking the subject from the side of mathematical 
problems. The same is true of harmony. Empirical methods, rely- 
ing upon the ear alone, are not adequate. Absolute rhythm, perfect 
harmony, is a matter of the relation of numbers; the method of its 
attainment is a mathematical problem. 12 

It has already been pointed out that Plato conceived of all the 
arts and sciences as resting upon a mathematical basis (see p. 29). 
Now, he holds further that the more the arts make use of the mathe- 
matical element, the more they partake of the nature of knowledge, 
and the more exact and scientific they become. In fact, the arts can 
be graded up and arranged in order on the basis of the extent to 
which they avail themselves of mathematics. His position in these 
respects can be illustrated from a passage in the Philebus, which I 
will summarize : 

In the productive or handicraft arts, one part is more akin to 
knowledge, and the other less ; one part may be regarded as 
pure and the other as impure. These may be separated out. If 
arithmetic, mensuration, and weighing be taken away from any 
art, that which remains will not be much. The rest will be only 
conjecture, and the better use of the senses which is given by experi- 
ence and practice, in addition to a certain power of guessing, which 
is commonly called art, and is perfected by attention and pains. 
Music, for instance, is full of this empiricism ; for sounds are har- 
monized, not by measure, but by skilful conjecture ; the music of the 
flute is always trying to guess the pitch of each vibrating note, and is 
therefore mixed up with much that is doubtful and has little which is 
certain. And the same will be found good of medicine and hus- 
bandry and piloting and generalship. The art of the builder, on the 
other hand, which uses a number of measures and- instruments, 
attains by their help to a greater degree of accuracy than the other 
arts. In shipbuilding and housebuilding, and in other branches of 
the art of carpentering, the builder has his rule, lathe, compass, line, 
and a most ingenious machine for straightening wood. These arts 
may be divided into two kinds — the arts which, like music, are less 
exact in their results, and those which, like carpentering, are more 
exact. 13 

12 Rep.. 7 : 529-31. 13 Philebus, 55-56. 



36 THE MATHEMATICAL ELEMENT IN PLATO'S PHILOSOPHY 

It is interesting to note that he goes on to sift out from these 
more exact arts that element — arithmetic, weighing, and measuring 
— on which their exactness depends, and to examine that with refer- 
ence to its cognitive and scientific character. Having got from 
mathematics the conception that the scientific character of a body of 
knowledge was dependent upon the power of exercising intellectual 
control through the principle of quantity, he went to work and 
applied that conception to mathematics itself. He demanded that 
mathematics be made scientific through the rigid application of its 
own principles. This latter point does not come out specifically in 
the passage here in the Philebus, but the principle of distinction 
involved m it is employed. He notes the wide difference between 
popular arithmetic and philosophical. In the former, reckoning is 
done by the use of unequal units ; " as, for example, two armies, two 
oxen, two very large things or two very small things." That is, 
units are used which are not determined on the basis of the prin- 
ciple of measuring. " The party who are opposed to them insist that 
every unit in ten thousand must be the same as every other unit." 
This same difference in accuracy exists between the art of men- 
suration which is used in building and philosophical geometry, also 
between the art of computation which is used in trading and exact 
calculation. The conclusion of the matter is that those arts which 
involve arithmetic and mensuration surpass all others, and that 
where these enter in their pure, or scientific, form there is infinite 
superiority in accuracy and truth. 14 Mathematics itself, then, if it is 
to be made scientific, must be based upon its own rigid principles. 
It is only when it is pursued in the spirit of the philosopher that it 
attains to its true cognitive function, that it reaches scientific 
knowledge. 15 

In concluding the discussion of the influence of mathematics 
upon Plato's conception of science, we may say two or three things 
by way of summary. He regards every art as having its scientific 
aspect, even the art of war, which we have not specifically dis- 
cussed. 10 This scientific aspect varies with the extent to which the 
art has been reduced to intellectual control through the use of 
mathematics. Also with reference to the sciences proper, they are 
to be deemed such by reason of the fact that they are bodies of 
knowledge the accuracy and validity of which are secured by the use 
of mathematical methods of procedure. 

u Philebus, 56-57- 15 Rep., 7 : 525-27. 10 See Rep., 7 : 522, 525, 526, 527. 



THE FORMULATION OF PHILOSOPHICAL PROBLEMS 37 

In Plato's cosmology, mathematics plays the same instrumental 
and intermediary part as in the arts. The cosmological problem is 
only a broadening out of the ontological one. We have already dis- 
cussed to some extent the ontological problem in relation to the 
analogy of the arts (see p. 29). There it was taken up not so much 
for its own sake as for the light which it threw upon the problem of 
knowledge and of ethics. There we saw that the arts are concerned 
with production, and that the intermediation between becoming and 
being was effected by the mathematical element. The Eleatic ele- 
ment of permanency was maintained as that which held in control 
the shifting stream of becoming. We now have to take up the same 
problem in its more general form. 

The Heraclitean-Eleatic opposition of becoming and being had 
already been resolved by philosophers who held to the doctrine of ele- 
ments and by others who postulated atoms, as the ultimate permanent 
and unchanging being. Generation and decay were accounted for on 
the basis of the integration and disintegration of complexes of these 
original elements. As in the case of production in the arts, so in 
the general case Plato saw something more in becoming than this. 
In the Phcedo he intimates his dissatisfaction with the explanation of 
generation and decay by separation and aggregation, by any prin- 
ciple of mere increase or decrease. He narrates how he had a youth- 
ful enthusiasm for the problem of generation and corruption (96), 
but that soon he got into all manner of difficulties. This is his 
account of the experience: 

There was a time when I thought that I understood the meaning of 
greater and less pretty well ; and when I saw a great man standing by a little 
one, I fancied that one was taller than the other by a head; or one horse 
would appear to be greater than another horse; and still more clearly did I 
seem to perceive that ten is two more than eight, and that two cubits are more 
than one, because two is the double of one. 

I should be far enough from imagining that I knew the cause of any of 
them, by heaven I should ; for I cannot satisfy myself that, when one is added 
to one, the one to which the addition is made becomes two, or that the two 
units added together make two by reason of the addition. I cannot under- 
stand how, when separated from the other, each of them was one and not two, 
and now when they are brought together, the mere juxtaposition or meeting 
of them should be the cause of their becoming two ; neither can I understand 
how the division of one is the way to make two ; for then a different cause 
would produce the same effect — as in the former instance the addition and 
juxtaposition of one to one was the cause of two, in this the separation and 



38 THE MATHEMATICAL ELEMENT IN PLATO'S PHILOSOPHY 

subtraction of one from the other would be the cause. Nor am I any longer 
satisfied that I understand the reason why one or anything else is either 
generated or destroyed or is at all, but I have in mind some confused notion 
of a new method, and can never admit the other. (96-97.) 

This is a very difficult passage to interpret. It is evident that 
from one point of view he is leading up to the problem of ends. He 
is feeling his way toward a final cause in the matter of the physical 
world. This is evident from his elaboration of the principle of the 
good in the passage immediately following. But within this whole 
problem of final cause there falls this one of generation and decay, 
the problem of becoming. Just as he is going to be dissatisfied with 
the causal explanation of the physical universe that has been given 
by Anaxagoras and others, so he is also dissatisfied with the explana- 
tion of the process of becoming that views it solely from the side of 
aggregation and juxtaposition. He takes up this problem in its most 
acute form — where it affects one's conception of relations. Plato 
seems to indicate, when he comes to resolve these contradictions 
(101), that they arise from the fact that the principle of explaining 
generation and decay — namely, that of aggregation and juxtaposi- 
tion on the one side, and disintegration and division on the other — 
was not a mathematical principle. If it had been, it would not have 
caused so much confusion and contradiction in the case of dealing 
with relations. Participation in number is an essential to the pro- 
cess of becoming. Whether we agree or not with the form in which 
Plato expresses this mathematical principle underlying the process, 
the essential point to note here is that he seems to be making for 
mathematics a function in the whole process of becoming. 

In the arts, order and determination were introduced into the 
process of production through the mathematical principles of number, 
measure, and weight. Passages in the Timczus would show that 
Plato had much the same conception of the whole cosmological pro- 
cess. We cannot go into the details of cosmology as outlined in the 
Timceiis, but only strike at a few of the most significant points for 
our purpose. The mathematical element is made very prominent. 
Two or three illustrations will be enough to exhibit the principle. 

We will start with his conception of the elements of the physical 
universe (53-57). He begins with the traditional four elements of 
earth, water, air, and fire. The old physical philosophers had 
explained becoming on the basis of the transformations of these 
elements, but they had no adequate technique of that process. Plato 



THE FORMULATION OF PHILOSOPHICAL PROBLEMS 39 

undertakes to explain the process by working out a technique for it 
on a mathematical basis. Each of these elements is itself made up 
of triangles, the particular mathematical principle employed being 
that of the construction of the regular solids — the regular pyramid, 
the octahedron, the icosahedron, and the cube. The cube is the form 
of the element earth ; the icosahedron, water ; the octahedron, air ; 
and the regular pyramid, fire. The stability, mobility, or decomposi- 
bility of these various elements is dependent upon their form and the 
relation of the triangles involved in their composition. These can all 
be expressed by a mathematical formula. The assumption of the 
truth of the account of the nature of the elements rests upon " a com- 
bination of probability with demonstration." The principles which 
are prior to the triangles " God only knows, and he of men who is 
the friend of God." Thus it will be seen that the triangles are not 
themselves regarded as ultimate, they are instrumental and inter- 
mediate. As in the arts, so here the mathematical element comes in 
as the factor of control, as that which makes technique possible, 
which gives the power of controlling means with reference to ends, 
of bringing forth being out of becoming, that is, making becoming 
not merely a random, ceaseless streaming, or process of " flux," but 
actually a process of Scorning. 

It is interesting to note also that the four elements themselves 
stand in a mathematical relation to one another (31-32). Between 
the densest and the rarest two means are inserted as a bond of 
union — fire is to air as air is to water as water is to earth. The 
creation of the universal world-soul was also conceived to have been 
by the taking of the elements of same, other, and essence, and com- 
bining them into a compound upon the basis of certain proportions 
with which Plato was familiar as lying at the basis of harmonies 
(35-36). The motions of the heavenly bodies, with all their diver- 
sity and complexity, were yet explained on the basis of a structure 
which rested upon mathematical principles (38-40). 

In the Laws 11 there is also an intimation that the processes of 
growth and decay involve mathematical principles. Plato speaks, in 
this connection, of the proportional distribution of motions, and he 
also uses a geometrical figure in describing the process of creation by 
increase from the first principle up to the body which is perceptible 
to sense. In another place 18 he defends himself against the charge 
of impiety for holding to a mathematical conception of the universe. 

17 Laws, 10 : 893-94. 1S Laws, 12 : 966-67. 



40 THE MATHEMATICAL ELEMENT IN PLATO S PHILOSOPHY 

Now, the upshot of all this is that when Plato used mathematical 
terms and mathematical figures in the discussion of cosmological 
questions, he did not use them as mere figures, or in a mystical sense 
either. He was carrying out, as best he could, in its application to 
the problems of the physical universe that conception of the signifi- 
cance of mathematics which he had learned from the arts as engaged 
in processes of production. Mathematical principles were as neces- 
sary to the creative activity of the deity as to the constructive 
activity of man. God and man were alike under necessity in this 
respect, 19 and that necessity is the necessity of intelligence, or mind. 20 

This interpretation of the place of mathematics in Plato's cos- 
mology, and in his thought generally wherever the question is one 
relating to becoming, genesis, or production, enables us to view as 
serious and intelligent some passages otherwise very perplexing. 
Among these are certain passages in the Laws, already referred to 
in another connection. 21 In these passages we can see an attempt 
to organize the state upon scientific principles, to introduce the 
element of rational control by the application of the principles of 
mathematics. Deference is paid to the mathematical relations 
involved in nature. In other cases, as in the use of the number 5,040, 
the particular number is probably not important, but the principle 
which it illustrates. 

A further curious instance of the same sort is found in the 
Republic.^" Here it is conceived that the perpetuity of the state 
could be indefinitely secured, provided the rulers had the wisdom to 
understand the mathematical law governing births ; for then they 
would have control of the birth of good and evil, and consequently 
could permit only those births which would be for the interest of the 
state. 

Further development of Plato's cosmology will be postponed for 
the present. It will be touched upon again after some discussion of 
method (see p. 91). Before turning to the problem of method, there 
remains a brief discussion of the relation of mathematics to Plato's 
idealism. 

Two distinctions have been brought out as necessitated by mathe- 
matics : one, on the side of content, or object of knowledge, namely, 
" the sensible " (to bparov) and " the intelligible (to vmjTov) ; the 

"Laws. 7:818. ™Laws, 12:967. 

21 See Laws, 5 : 737, 738 ; 745-47 ! 6 : 771 ; and cf. 6 : 756. 

22 Rep., 8:545-47- 



THE FORMULATION OF PHILOSOPHICAL PROBLEMS 41 

other, on the side of faculty, mental activity, or process, namely, 
"sense" ($6£a) or opinion and "intellect" (vo^o-is) (see p. 26). A 
distinction of value also comes in (see p. 28) which tends toward 
idealism, namely, the minimizing of sense and the exaltation of rea- 
son. It is a familiar fact of Plato's philosophy that he exalts that 
which comes through mental function, whether reason or direct intui- 
tion of the soul, to the highest rank, and regards nothing as partaking 
of scientific character and worthy to be called knowledge which 
comes through sense alone or is empirically derived. 23 This transi- 
tion to the idealistic point of view is equally bound up in the logical 
a priori point of view and in the mathematical. It is only in the 
interaction of the two points of view that the mathematical element 
gets its deepest significance. It is because the idealistic interpretation 
of things appeals so strongly to Plato as the direction in which to look 
for the solution of his philosophic problems that he is charmed and 
fascinated by the idealism of mathematics and so eagerly points it out 
and snatches at it. 

The idealism of mathematics clarifies, illumines, gives force and 
content to Plato's idealistic demand. This comes out both on the 
process and the content side of the subject. Take the following 
statements as evidence : 

Masters of the art of arithmetic are concerned with those num- 
bers which can only be realized in thought, necessitating the use of 
the pure intelligence in the attainment of the pure truth. 24 Arith- 
metic must be studied until the nature of number is seen with the 
mind only. 25 "The art of measurement would do away with the 
effect of appearances, and, showing the truth, would fain teach the 
soul at last to find rest in the truth." 26 Arithmetic compels the soul 
to reason about abstract number, and rebels against the introduction 
of visible and tangible objects into the argument. 27 Geometricians, 
" although they make use of visible forms and reason about them, are 
thinking, not of these, but of the ideals which they resemble ; not of 
the figures which they draw, but of the absolute square and the abso- 
lute diameter and so on — the forms which they draw or make, and 
which have shadows and reflections of their own in the water, are 
converted by them into images, but they are really seeking to behold 
the things themselves." 28 

To return to the discussion, these passages show that in the mind 

23 Rep., 6:510; 7 : S27, S29, 530-31, 523. 25 Rep., 7 ' 525. " Rep., 7 : 525. 
21 Rep., 7 : 525-26. 2a Protag., 356. 28 Rep., 6 : 510. 



42 THE MATHEMATICAL ELEMENT IN PLATO S PHILOSOPHY 

of Plato mathematics has continually the double process-result 
idealistic function. Its superiority as knowledge lies in the fact that 
it is most largely free from the sense-element. Objects of sense are 
to be distrusted. On the side of process, mathematics is engaged in 
getting away from them. It is exercising the mind, leading the soul 
away from the realm of sense. Although the mathematician may 
start with the data of sense as suggestive of his problem, these data 
are only the images of the absolute realities lying behind them, and 
his problem becomes truly mathematical only when he has made the 
transition to data that are purely ideal. On the side of content or 
result, mathematics furnishes to Plato the most conspicuous instance 
of a science which deals with absolute realities. Through starting 
with data which have been stripped by abstraction of their sense- 
elements and then have been ideally transformed, and then drawing 
conclusions by processes wholly rational or intuitive, the results 
attained are absolute, unchanging, necessary. They serve as the 
idealistic model for all scientific knowledge. Any subject of study in 
order to become scientific must, according to Plato, yield itself to this 
movement. This is brought out in his discussion of astronomy and 
harmony in the Republic. 29 

Both astronomy and harmony are very rich in the sense-element, 
but Plato feels that so long as this is not transcended and left behind 
we do not get the realities involved in them. These subjects must 
be made rational rather than empirical, and they become rational 
only by being made mathematical. 

Plato ridicules the idea that star-gazing is astronomy. This is 
" seeking to learn some particular of sense," and " nothing of that 
sort is matter of science." The spangled heavens may be glorious 
and beautiful to the sense of sight, but the geometrician "would 
never dream that in them he could find the true equal, or the true 
double, or the truth of any other proposition." In the study of 
astronomy the gift of reason must be made use of, the mathematical 
method must be applied, the proper procedure in the solution of 
problems. That which is eternal and subject to no variation must 
be sought ; but nothing that is material and visible can be eternal 
and subject to no deviation. 30 The empirical study of harmony is 
also held up to the same sort of ridicule, and for the same reason — 
that sense-perception is placed before reason and that absolute 
realities arc not attained. Failure here, too, is due to not applying 

28 Rep., 7 : 529-30 and 530-31- "° Rep., 7 '• 529-30. 



THE FORMULATION OF PHILOSOPHICAL PROBLEMS 43 

the mathematical method. The empirical students "set their ears 
before their understanding." Even the Pythagoreans " are in error, 
like the astronomers ; they investigate the numbers of the harmonies 
that are heard, but they never attain to problems." 31 

These two discussions — one on astronomy, the other on harmony 
— both illustrate very strikingly the distinction of value for knowl- 
edge which Plato makes between the sense-perception element and 
the intellectual element, and how through mathematical procedure 
this distinction leads over into idealism. Plato demands of knowl- 
edge that which is absolute, eternal, invariable. In the fields of 
astronomy and harmony he finds this demand met only through 
mathematical procedure. The truth, the reality which cannot be 
found on the side of sense-perception, can be found in the results of 
the rationalistic mathematical process. 

In always playing this double part of going through processes 
that lead over into the realm of ideas and of furnishing results that 
belong to that realm, mathematics, as it were, both furnishes the 
stimulus to idealism and is idealism. Certainly in the building up of 
Plato's idealistic philosophy mathematics, though not the only factor, 
is a very important one. The idealism of mathematics furnishes him 
with one of the strongest arguments by analogy for a universal 
idealism. Just as the ultimate reality with which the mathematician 
deals does not spring out of data of sense by any empirical process, 
but is both in respect to its real data and in respect to its final results 
something absolute and transcending sense-perception ; so with the 
ultimate reality behind all phenomena, it is the ideas, something in 
harmony with the rational principle of the soul, not subject to change, 
to the flux of the imagery of sense-perception. Only, in mathematics 
the process by which the material of sense is transcended and ideas 
are reached is capable of being exhibited, whereas in other realms it 
is not. This feature of mathematics is one key to the understanding 
of the importance which Plato attaches to mathematical training as a 
preparation for the study of philosophy. Without such training, on 
the one hand, the problem of philosophy, the problem of being or 
essence, cannot be adequately understood ; nor, on the other hand, a 
suggestion as to the process, or technique, of its solution arise. 

The problem of philosophy for Plato is to know true being. The 
function of the philosopher is to find through reason the absolute 
truth, the eternal being, lying behind and controlling all the phe- 

31 Rep., 7 : 530-31 ; cf. Phileb., 55-56. 



44 THE MATHEMATICAL ELEMENT IN PLATO S PHILOSOPHY 

nomena of sense. But on logical a priori grounds this knowledge 
cannot come through the channel of sense-perception ; for the senses 
are inadequate. Ordinarily " the eye of the soul is literally buried in 
an outlandish slough of sense. 32 Some preliminary training is needed 
in the idealistic process before the soul can rise to that height of 
freedom and power and self-control where she can gaze on absolute 
being and attain true knowledge. Mathematics serves the function 
of giving that training ; here intellect has found ultimate realities 
which are abiding, absolute, necessary, ideal. 

The philosopher must be an arithmetician, studying the subject 
until the nature of number is seen with the mind only, and for the 
sake of the soul herself ; this will be the easiest way for her to pass 
from becoming to truth and being. 33 The true use of number is 
simply to draw the soul toward being. 34 Geometry also gives the 
same valuable idealistic training. It tends to make more easy the 
vision of the idea of the good, compelling us to view being and not 
becoming only. Its real object is knowledge, and the knowledge at 
which it aims is knowledge of the eternal and not of aught perishing 
and transient. Then geometry will draw the soul toward truth and 
create the spirit of philosophy. 35 The study of harmony is useful 
to the same end, if it be made mathematical and be studied " with a 
view to the beautiful and good." 36 

It is thus seen that Plato feels that the mind which has accus- 
tomed itself in the realm of mathematics to make the transition from 
the exercise of the senses to that of the intellect, and has acquired 
the power of abstraction and of centering its attention upon purely 
ideal elements, is the only mind fit to philosophize. The training 
of mathematics is positive, direct, and necessary in preparing the 
mind for that point of view which seeks the ultimate reality of all 
things in ideas as over against the products of sense-perception. 
While Plato seems to reserve specifically to dialectic the power to 
reveal the absolute truth, the ultimate reality, yet he feels that it can 
reveal this only to one who is a disciple of the mathematical sciences, 
which are used as " handmaids and helpers " in the work of uplifting 
the soul. 37 

Before leaving this discussion of the idealism of mathematics it 
will be necessary to take account of an important passage in the 

32 Rep., 7 : 533. M See Rep., 7 : 526-27. 

".,7:525. m Rep.,7'.s*i. 

:i Rep., 7 : 523 ; see also 7 : 521-23 and 523-25. 37 Rep., 7 ' 533- 



THE FORMULATION OF PHILOSOPHICAL PROBLEMS 45 

Republic — the famous figure of the divided line. 38 From this pas- 
sage it would appear that Plato does not place mathematical notions 
on a level with the ideas. In this passage there is a discussion of the 
stages of knowledge, or, in ontological terms, the degrees of being. 
First, two main divisions are made : Opinion (So£a), the lower, 
which is concerned with the visible world (to oparov) and has to do 
with becoming (ycVeo-is); and Intelligence or Thought (vo'170-is), 
which is concerned with the intelligible world (to votjtov) and has to 
do with Being (oka). Opinion (Sd£a) is itself divided into two 
stages : Conjecture (eiVao-ta), which has to do with images (e'Uoves) 
in the nature of shadows and reflections ; and Belief (ttlo-tls), which 
has to do with things — the animals which we see and everything 
that grows or is made. Intelligence is also divided into two parts : 
Understanding (Siavoia), which Plato makes clear, works with 
images of things, but to which he does not make clear that there is 
any corresponding distinct object of knowledge or being; and 
Reason (vovs), which has to do with the Idea (iSc'a) or eternal Being. 
Schematizing this, it would be something as follows, without attach- 
ing any significance to the length of the lines used. 

( 56|a v6t]cn$ 

ies X 



56|a 
Faculties 

A / eiKacria I 7r/<rm 



didvoia I vovs 



y ( eticoves I things fMa6r}/j.aTLKd(?) | idtcu 

J j rb 6par6v rb vbiyrov 

On the side of faculty, it is clear that there are four divisions, 
of which the third is Understanding (Stavota), and that this third 
faculty is the one concerned with mathematical thinking. 39 Now, in 
making a distinction on the side of process between mathematical 
thinking and reason (oWoia and vovs), it is a question whether 
Plato wishes to make a distinction in the nature of their objects — 
mathematical truths (/xaflry/xaTiKa) and Ideas (iSe'ai). Milhaud shows 
at considerable length that such an interpretation does not hold. He 
points out the fact that the fundamental line of division with Plato 
is the twofold one, that according to this line of division mathematical 
notions and ideas belong together as not different in essence, and 
that it is more in harmony with the spirit and usage of Plato to 
identify than to separate them. 40 This proof of Milhaud would have 

38 Rep., 6 : 508 ff . 39 Rep., 6 : 5 1 1 . 

40 In Milhaud see especially pp. 242, 244, 263, 267, 270-72, 274 ff., 277-79. 



46 THE MATHEMATICAL ELEMENT IN PLATO'S PHILOSOPHY 

been still clearer if he had observed carefully the distinction which 
Plato makes between faculty, or method, and object. On the side of 
method (a point which will be discussed further later on, p. 84) the 
distinction between mathematical thinking (Understanding, or 
Siavoia) and pure Reason (vovs) is clear; on the side of object it 
is not clear ; hence the ambiguity, the problem, the apparent incon- 
sistency of Plato. On the side of process four divisions in all, on 
the side of object only three, or at least the distinction between a 
third and a fourth merely formal. That the distinction on the side 
of object between mathematical essences and Ideas is formal and 
apparent rather than real 41 in the mind of Plato, take for proof this 
passage on the identity of their nature right from the midst of the 
discussion of the different divisions : 

And do you not know that, although they [the mathematicians] make use of 
their visible forms and reason about them, they are thinking not of these, but 
of the ideals which they resemble ; not of the figures which they draw, but of 
the absolute square and the absolute diameter, and so on — the forms which 
they draw or make, and which have shadows and reflections in water of their 
own, are converted by them into images, but they are really seeking to behold 
the things themselves, which can only be seen with the eye of the mind? 42 

The mathematician may start with data of sense, but these he 
ideally transforms and transcends, and comes out with absolute and 
ideal results. 

41 1 am speaking in ontological terms here. On the side of cognition, I hold 
that the distinction is a real one in the mind of Plato, but not absolute. See p. 87. 
i2 Rep., 6: 510. 



CHAPTER III. 

METHOD, OR THE TECHNIQUE OF INVESTIGATION. 
PLATO'S INTEREST IN METHOD. 

It has already been suggested that Plato is as much interested in 
mental processes as in mental results (see p. 19). I hold that we do 
not get the full significance of Plato's dialogues when we view them 
from the side of literature alone, nor from the side of their philo- 
sophical content alone. Either, or both together, is a mistaken view. 
In Plato we have a most remarkable exhibition of a man who reveals 
the psychological and logical processes by which he reached his con- 
clusions. This he did with such literary skill and power, and the 
problems with which he dealt were of such magnitude, that we are 
apt to get lost either in the literary side of his productions or in the 
realm of their ideas. Plato might have given us only the results of 
his search ; but, whether purposely or otherwise, in many of his 
dialogues he has given more of the method than of the actual result. 
This quite likely is not an unconscious matter of chance with him. 
He was a master of the pedagogical method of stimulating and 
awakening interest in the problems which he discussed before going 
on to give his own views. He recognizes the significance of the 
principle of " shock " which comes from conflict and contradiction. 

Do you see, Meno, what advances he has made in his power of recollec- 
tion? He did not know at first, and he does not know now, what is the side 
of a figure of eight feet; but then he thought he knew, and answered con- 
fidently as if he knew, and had no difficulty; now he has a difficulty, and 
neither knows nor fancies that he knows. Is he not better off in knowing 
his ignorance ? If we have made him doubt, and given him the " torpedo 
shock," have we done him any harm? We have certainly, as it would seem, 
assisted him in some degree to the discovery of the truth; and now he will 
wish to remedy his ignorance, but then he would have been willing to tell all 
the world again and again that the double space should have a double side. 
But do you suppose that he would ever have inquired into or learned what he 
fancied that he knew, though he was really ignorant of it, until he had fallen 
into perplexity under the idea that he did not know, and had desired to know. 1 

It is very noticeable that often before getting down to the most 

1 Meno, 84. 

47 



48 THE MATHEMATICAL ELEMENT IN PLATO'S PHILOSOPHY 

serious discussion of a subject he plays, as it were, with the ideas of 
his interlocutors. This, too, he sometimes does quite eristically, as 
shamelessly from the point of view of logical principles as any 
Sophist, but after all in a manner which is most often on a level with 
the logical processes of his respondents or opponents, and which is 
calculated to show by the contradictions into which he leads them the 
inadequacy and shallowness of their thinking. This accomplishes 
two things : ( I ) it stimulates an interest in the real rather than the 
superficial aspects of problems ; (2) it prepares the way for the con- 
sideration of better methods and the introduction and appreciation 
of more rigid logical processes. 2 

We do injustice to Plato if we fail to see that he has a genuine 
pedagogical interest in method, and so come to overlook the fact that 
there are many long digressions in his dialogues which are not to be 
regarded as significant so much for their content as for the illustra- 
tion and explanation of some method or principle of procedure. As 
a sample of such a digression take the discussion of the principle of 
contradiction in Thecetetus, 155, and in Republic, 4:436-39. Large 
portions of the Republic, the Sophist, the Statesman, and the Par- 
menides come under this same head. The larger part of the Republic, 
while undoubtedly expressing sociological ideas which Plato wished 
to discuss, is from another point of view explicitly a point in method- 
ology. From this point of view the ideal state is only a construction 
devised for the purpose of working out a psychological analysis of 
justice. The first book has failed to issue in a definition of justice. A 
suggestion is then made to proceed to another method. 

Seeing then, I said, that we are no great wits, I think we had better 
adopt a method which I may illustrate thus : Suppose that a short-sighted 
person had been asked by someone to read small letters at a distance; and 
it occurred to someone else that they might be found in another place which 
was larger and in which the letters were larger — if they were the same and 
he could read the larger letters first, and then proceed to the lesser — this 
would have been thought a rare piece of good fortune. 3 

The ideal state is nothing but the letters writ larger in a larger 
place. The analysis of justice here gives the key to the analysis of 
justice in the individual. 4 

This same principle of analyzing a plain and simpler case for the 
sake of Its application to the more complex and obscure is again 
enunciated in the Sophist (218) and in the Statesman (286). So 

1 Cf. Soph., 230. 3 Rep., 2 : 368. ■* See Rep., 4 ■ 434 &• 



METHOD, OR THE TECHNIQUE OF INVESTIGATION 49 

clearcut and significant is the statement in the Sophist that it may be 
well to quote it : 

Now the tribe of the Sophists which we are investigating is not easily 
caught or defined; and the world has long ago agreed that if great subjects 
are to be adequately treated, they must be studied in the lesser and easier 
instances of them before we proceed to the greatest of all. And as I know 
that the tribe of Sophists is troublesome and hard to be caught, I should 
recommend that we practice beforehand the method which is to be applied to 
him on some simpler and smaller thing. 5 

The case of the angler is taken up in great detail, illustrating 
definition through a long process of division of species. When this 
is complete, then " following this pattern " they " endeavor to find out 
what a Sophist is." 6 Is it too much to think that not only this illus- 
tration, but also the whole dialogue, has for one of its great purposes 
an exposition of method — the method of logical division whereby 
mutually exclusive alternatives may be secured ? (See pp. 48, 58, 61.) 

In the Statesman there is again the explicit recognition of digres- 
sion from the main line of argument for the sake of methodological 
reasons. Take the passage where the art of weaving has been 
analyzed by the process of division and has led up to the discussion of 
the art of measurement. Now that this discussion is completed the 
proposition is made "to consider another question, which concerns 
not only this argument, but the conduct of such arguments in gen- 
eral/' Attention is called to the fact that in teaching a child the 
letters which go to make up a word the aim is not merely to improve 
his grammatical knowledge of that particular word, but of all words. 
So, in the case of the analysis of the art of weaving, it has not been 
done for its own sake, but for the sake of the training in giving and 
accepting a rational account of things. 7 

Exposition of method and training in method is one of the motifs 
of the dialogues " of search." 8 Even positions are taken in which 
the speaker does not necessarily believe, for the sake of the argument. 
Thrasymachus says : " I may be in earnest or not, but what is that 
to you? — to refute the argument is your business." 9 Glaucon main- 
tains the cause of injustice, though confessedly not believing it, in 
order that he may see how the position can be refuted. 10 In the 
Gorgias (462) Socrates gives Polus a lesson in the method of argu- 

6 Soph., 218. 6 Soph., 218-21. 7 Stat., 285-86. 

8 I use this terra to apply more widely than to the minor dialogues alone. 

s Rep., 1:349. 10 Rep., 2:358. 



50 THE MATHEMATICAL ELEMENT IN PLATO S PHILOSOPHY 

mentation. In the Phcedo (100 ff.) there is quite a long digression 
on method. The Parmenides is from one point of view, taken as a 
whole, an exposition of method. When Socrates is involved in cer- 
tain difficulties regarding the Ideas, Parmenides explains that this 
arises out of his attempting to define the beautiful, the just, the good, 
and the ideas generally, without sufficient previous training. The 
special lack in his training is pointed out, and it is indicated wherein 
he could get better training. Parmenides is asked to give an illus- 
tration of the method which he has indicated. This he proceeds 
reluctantly to do. On the side of form, the dialogue Parmenides is 
the outcome. 11 

PLATO'S DOGMATISM. 12 

It may seem a little inappropriate at first thought to discuss dog- 
matism under the general head of method. But there seems to be 
no better place for its discussion in this book without involving 
repetitions. The dogmatism of Plato has a bearing on method, even 
if it cannot be said to be a part of method. The term " dogmatic " 
is often used as a term of reproach. Plato was a dogmatist in the 
better sense of the word. He was in full sympathy with the process 
of investigation; he longed for the truth above all things. But he 
also believed that the process of search was capable of resulting in 
valid conclusions. His confidence in knowledge is well illustrated in 
a passage in the Meno, where the following declaration is put in the 
mouth of Socrates : 

Some things I have said of which I am not altogether confident. But that 
we shall be better and braver and less helpless if we think that we ought to 
inquire than we should have been if we indulged in the idle fancy that there 
was no knowing and no use in seeking to know; that is a theme upon which 
I am ready to fight in word and in deed to the utmost of my power. 13 

Compare this with the statement of Meno, 98 : 

That knowledge differs from true opinion is no matter of conjecture with 
me. There are not many things which I profess to know, but this is most 
certainly one of them}* 

How significant is this language which describes the assumption 
that there is no knowledge as an " idle fancy," and the statement 
" that knowledge differs from true opinion is no matter of conjecture 
with me, but one of the things which I most certainly know ! " 

11 farm., 135-36. 

u I use dogmatism as opposed to skepticism, not as opposed to criticism. 

13 Meno, 86. M See also Phccdo, 99, 100, and 107. 



METHOD, OR THE TECHNIQUE OF INVESTIGATION 5 1 

On a priori grounds Plato was a dogmatist. It seems to be 
claiming too much, as Milhaud does, to trace this dogmatism wholly 
to the mathematical influence. But given a dogmatic tendency and 
problems so vital that they demand a positive solution, then we might 
expect that any branch of study which lent itself in support of that 
dogmatic tendency would be of great significance in clarifying and 
strengthening it. In the interaction of the dogmatic tendency of 
Plato's mind, as it swept out on the a priori movement of thought in 
the search for truth with the dogmatic element in mathematics, this 
mathematical element received a significance for him which it other- 
wise would not have had. Plato's unswerving confidence in knowl- 
edge coincides with what we should expect from the mathematician. 
The conclusions of the mathematician strike the mind with all the 
force of necessity. They seem absolute, irrefutable, independent of 
the fluctuations of the sense manifold. Mathematics is a realm where 
we actually have knowledge. Descartes, Locke, and Hume felt this. 
Kant made the a priori synthetic judgments of mathematics and 
pure physics the starting-point of his thought. There seems to be 
good reason for believing that mathematics is a fundamental factor, 
though not the only one, in Plato's dogmatism. The dogmatism of 
mathematics must have been a powerful support and stimulus at all 
times to that natural dogmatism which made Plato expect to find a 
secure basis for knowledge because on logical grounds ethics 
demanded it. 

Plato's dogmatism, I have said, has a bearing on his method 
which justifies its treatment at this point. It gives his processes of 
thought a firm basis of reaction ; it is the fulcrum, as it were, upon 
which his method turns. It gives stimulus and emphasis to the 
process of search. In the Phcedo he issues a warning not to distrust 
arguments merely because they fail — not to hate ideas, but to search 
more diligently. 16 It is from this point of view that Plato lays so 
much stress on the principle of contradiction. If knowledge is pos- 
sible, and not merely a highly relative and fleeting product of the 
senses, then any procedure which would reduce knowledge to the 
test of a standard which is not absolute is inherently wrong. The 
possibility of knowledge is the rock against which all else breaks. 
There is a significant passage in the Sophist bearing on this point. 

And surely contend we must in every possible way against him who would 
15 Phcedo, 89-91; cf. 99-100. 



52 THE MATHEMATICAL ELEMENT IN PLATO S PHILOSOPHY 

annihilate knowledge and reason and mind, and yet ventures to speak con- 
fidently about anything. 16 

It is the dogmatic attitude which gives vitality to the principle of 
contradiction, so that we are not content to rest with contradictory 
conclusions, but insist on continuing the search with the feeling that 
there must have been something wrong with the method employed 
or with the premises used. The principle of contradiction occupies a 
central position in Plato's thought. Let me give two passages from 
the Gorgias in illustration of his feeling on this point : 

I would rather that my lyre should be inharmonious, and that there should 
be no music in the chorus which I provided; aye, or that the whole world 
should be at odds with me, and oppose me, rather than that I myself should 
be at odds with myself, and contradict myself. 17 

For my position has always been, that I myself am ignorant how these 
things are, but that I have never met anyone who could say otherwise, any 
more than you can, and not appear ridiculous. 18 

MATHEMATICS AND METHOD 

In connection with the dogmatic attitude of mind and its relation 
to the principle of contradiction, it has been pointed out that mathe- 
matics plays a part in determining Plato's method. This phase of 
the mathematical influence now needs to be worked out along other 
lines. The significance of mathematics as a factor in determining 
Plato's method is usually overlooked or underestimated. Doubtless 
one reason for this is that the influence of Socrates upon Plato has 
been regarded as decisive. This is an easy inference to make ; for 
Plato adopts the question-and-answer form of discussion, which 
appears on the surface to be Socratic. Again, he is much engaged in 
the Socratic task of analyzing concepts. We do not deny that Plato's 
method is Socratic in its early stages, and that it always retains 
Socratic elements. But the ethical problem took such a form with 
Plato that he was obliged to transcend his teacher and seek a more 
rationalistic method of procedure. This he worked out by com- 
bining with the Socratic principles the logical a priori demand and 
certain principles of mathematical reasoning. The outcome was a 
method which in all that is most essential to it is wholly un-Socratic. 
Let us take up in more detail some of the leading factors which 
entered into the Platonic reorganization of method. 

In the first place, there was a marked difference in the way in 

"Soph., 249. " G orgies, 482. 1S Gorgias, 509. 



' METHOD, OR THE TECHNIQUE OF INVESTIGATION 53 

which Socrates and Plato approached the ethical problem. We may 
regard the ethical interest as fundamental with both of them, but in 
different ways. After the destructive criticism of the materialistic 
Protagorean school, which did away with the ethical sanctions inher- 
ent in custom and the traditional polytheistic religion, these two men 
sought to save Athenian society by refounding ethics upon new and 
more secure moral sanctions. The ethical concepts underwent search- 
ing empirical and inductive examination and investigation by 
Socrates, with a view to showing their validity in actual experience. 
He was not so much concerned with the problem of the possibility of 
ethical standards, or universals, as with the fact of their existence as 
factors real and operative in human life. This he offsets against 
Protagorean theory. But Plato goes a step farther and meets theory 
with theory. He has a living interest in theoretical problems. This 
is decidedly a non-Socratic impulse. It undoubtedly grows out of 
the conflicting conclusions of the philosophical systems with which 
he was familiar. To Socrates these conflicts indicated the folly of 
speculative philosophy and the necessity of abandoning it and of 
confining himself to practical problems. But to Plato, under the 
influence of the dogmatic temperament and a mathematical training, 
these conflicts could not lead to skepticism. They were to him the 
stimulus to further investigation and final solution. 

In accordance with these theoretical interests, Plato came to the 
logical demand for the distinction between the senses and the 
intellect, and hence for universals recognized by the mind alone and 
untainted by any sense-elements. At whatever time this logical 
demand became operative in his thought, whether with vague or with 
clear consciousness, whether alone or in conjunction and interaction 
with the mathematical line of thought, there must have begun a feel- 
ing of dissatisfaction with the Socratic universals. They were 
grounded ultimately in the world of sense-perception. They were 
derived empirically, through the analysis of the opinions of men, and 
could have no content except such as they found there. So long as 
they were tainted with the fleeting and inadequate sense-element, and 
not grounded in reason alone, they could not meet the supreme test of 
knowledge in accordance with the logical a priori demand. 

In connection with this movement of thought it is significant that 
Plato was a student of mathematics. Socrates held theoretical mathe- 
matics (together with other theoretical subjects) in contempt. In so 
far as he did care for mathematics, it was solely on the practical side, 



54 THE MATHEMATICAL ELEMENT IN PLATO S PHILOSOPHY 

and that, too, in a very limited field. Plato, on the contrary, held 
mathematics in the highest esteem, especially theoretical mathematics, 
insisting everywhere and always on the superiority of abstract, or 
pure, mathematics in respect to scientific accuracy and truth. This 
is a decisive, non-negligible factor in the determination of his method. 
On the basis of the logical a priori demand, the Socratic universals 
are seen to be unsatisfactory. Mathematics led to the same conclu- 
sion. But more important than its coinciding with the other move- 
ment of thought in this respect is its clarification and grounding of 
that position. In mathematics knowledge is unquestionably found, 
and in this field its characteristics are plainly to be discerned. They 
are clearness, certainty, necessity, universality, rationality, and espe- 
cially rigor of derivation. These then are the demands which con- 
cepts must meet if they are to be called knowledge. On a priori 
grounds knowledge must be wholly rational, free from the taint of 
the sense-element. This demand receives specific content from the 
realm of mathematics. In that realm we see clearly what the condi- 
tions are which knowledge must satisfy. These are the very same 
demands which must be met in every other field. When insight into 
the nature of mathematics has made these plain, the concepts of Plato 
can no longer be the inductively derived concepts of Socrates. Nor 
can they be these concepts merely hypostatized, if we mean by 
hypostatization the taking of them as they are, giving them an object- 
ive existence, and setting them up arbitrarily as independent of the 
phenomenal world. On this basis they would no more meet the 
demands of knowledge than before their hypostatization. Besides, it 
would be a reversal of the order which Plato gives to them ; for it 
would be deriving the ideal from the phenomenal, instead of the 
phenomenal being a copy of the ideal, the former serving only as a 
stimulus to the search for the ideal. The ideality of the Platonic 
universals, if mathematical notions may be taken as an illustration of 
them, is not an ideality arbitrarily thrust upon them merely to meet 
an a priori logical demand. It is an ideality inherent in the very 
nature of knowledge. In mathematics, the demands of knowledge 
are met because of the ideality of its realities. The method by which 
such realities are attained in mathematics should be suggestive of the 
conditions which must be fulfilled in order to attain them in other 
realms. 

The procedure of mathematics is especially marked by the fact 
that it starts out with data which it receives through intuition 



METHOD, OR THE TECHNIQUE OF INVESTIGATION 55 

(direct mental vision) or through intellectual determination, and 
having found such data it then works under such rigid rules of infer- 
ence as to exclude any consequences which might creep in through- 
premises inadvertently brought in from the outside. The whole pro- 
cess is intellectually controlled in such a manner as to exclude the 
introduction of empirical elements or empirical processes at any stage 
of the procedure. There is abundant evidence that the essential 
features of this process appealed strongly to Plato. 

Plato's appreciation of the intuitive element in mathematical 
method has already been brought out (see p. 10) by the incident cited 
from the Meno. 19 It was the a priori character of mathematics, the 
non-empirical origin of its truth, that impelled Plato to evolve the 
doctrine of recollection. At least, the proof that he gives of it is 
based most largely on the intuitive element in geometry. It seems 
altogether probable that Plato's doctrine of recollection 20 and his 
doctrine of philosophic love 21 vaguely play the part in his philosophy 
that a priori synthetic judgments consciously do later in Kant's 
philosophy — the undoubted existence of mathematical truths settling 
dogmatically the question of the possibility of knowledge, these doc- 
trines serving the function of showing how it is possible, through the 
acquisition of non-empirical elements with which to start. 

In addition to intuitive data, there are data intellectually deter- 
mined. Plato regards careful definition as important in any method 
of philosophic procedure. 22 In all his dialogues of search it may be 
noticed that every new discussion is prefaced by a skirmish as to the 
meaning of terms, or at least a basis of agreement. 23 Some dialogues 
seem to be almost wholly given up to the process of sifting tentative 
definitions until a satisfactory one — i. e., a rigorous one — can be 
found ; or, if this is impossible, the search is abandoned with the clear 
consciousness that the definitions given are inadequate. Plato's con- 
ception of definition is the outcome of the interaction of the twofold 
movement of thought already described — the logical a priori and the 
mathematical. Milhaud claims too much for the mathematical ele- 
ment in the elevation of Plato's conception of definition above the 
Socratic point of view. 

Plato views definition as something more than the matter of a 

19 Meno, 81-86; see also Phcedo, 73 ff. 

20 Meno, 81 ff. ; Phcedo, 73 ff. ; Phcedrus, 249 ff. 

21 Phcdrus, 249-53. m See Phcedrus, 237. a Cf. Phileb., 20. 



56 THE MATHEMATICAL ELEMENT IN PLATO'S PHILOSOPHY 

name. 24 Definition is concerned with the general idea, 25 with the 
nature of a tiling rather than any description of its qualities. 26 It 
must not be expressed in terms of itself, 27 nor in terms as yet 
unexplained or unadmitted. 28 This latter point he takes from mathe- 
matics. Enumeration is inadequate as a principle of definition. This 
is worked out in non-mathematical instances, such as the definition of 
clay, as well as in the case of mathematical terms. The trouble with 
the definition which names kinds is that it does not reveal the nature 
of a thing in the abstract. 29 Definition must distinguish the class of 
things referred to from all others. This requisite is worked out 
through starting with an illustration from mathematics, followed up 
by several illustrations not mathematical, with a return to mathe- 
matics. 30 Both in the point made with reference to enumeration and 
in this latter point it is evident that Plato might have come to his 
conception of definition entirely apart from mathematical considera- 
tions. But that mathematics had for him a significance in this respect 
that should not be ignored seems evident when we observe how fre- 
quently he turns to mathematics for illustrations of principles in 
definition, and especially how he lingers over these illustrations and 
develops their significance. That Plato's conception of definition is 
very different from the Socratic conception of it as the expression of 
the common quality 31 is most clearly seen from a passage in the 
Phadrus. Two important principles are there enunciated: (1) The 
comprehension of scattered particulars in one idea. This is illus- 
trated in the definition of love given, " which, whether true or false, 
certainly gives clearness and consistency to the discourse. The 
speaker should define his several notions and so make his meaning 
clear.'' (2) Natural division into species. Socrates is made to say 
that he is " a great lover of these processes of division and generaliza- 
tion ; they help him to speak and to think." 32 These two principles 
both point in the direction of intellectual control of the sense mani- 
fold through logical processes centering in an idea. In so far as it 
does point that way, it is in harmony with a characteristic of mathe- 
matical definition. Here is a realm where the very essence of defini- 
tion is intellectual determination, comprehending the scattered 

24 Soph., 218. ■ Phccdo, 105. 

" Eulhyphro, 6. M Meno, 79. 

m Gorg., 448 ff. 28 Thecet., 146-48 ; see also Meno, 71-76. 

30 Gorg., 451-53- 3l Laches, 192. -' Phccdrus, 265-66; cf. Soph., 253. 



METHOD, OR THE TECHNIQUE OF INVESTIGATION $? 

particulars in one idea by the free activity of the mind in mental 
construction, rather than selecting some common quality that is dis- 
covered to run through and attach to all the particulars and defining 
on the very doubtful (for Plato because sensational) basis of that. 
Plato feels that what philosophic method needs is what mathematics 
is actually coming to have in his day — clear, unambiguous, intel- 
lectually determined definitions ; and he is very fond of showing up 
the absurd conclusions which can be derived from accepted premises 
merely by playing upon slight ambiguities in the meaning of terms 
employed. No rigorous conclusions in philosophy, he feels, can be 
obtained except as in mathematics ; i. e., by employing a method of 
procedure which shall continuously work within the limits of con- 
cepts which are either intuitive in character or intellectually deter- 
mined, and by means of rational processes which are carefully 
guarded at every step. 

THE METHOD OF ANALYSIS. 

When we go down deeper still into the very form and structure 
of Plato's argument, we find, in so far as the use of the method of 
analysis is conclusive, that his thought is even more profoundly 
modified by the influence of mathematics than appears from those 
passages where mathematical terms or mathematical usage enter 
expressly into the discussion. We find whole dialogues in the dis- 
cursive and constructive period of his thinking where the movement 
of his thought takes the form of the mathematical method of analy- 
sis. He talks a great deal about dialectic, but he actually uses the 
method of reductio ad absurdum. This method, in its earlier and 
less rigid state of development, connects itself so closely in form 
with Socratic analysis that the student may not easily notice the 
transition from one to the other, wide apart as they are when the 
transition is once made. But before that which is distinctive in 
Plato's analysis can be made plain, it will be necessary to discuss 
mathematical analysis more in detail. 

In the time of Plato the method of analysis in mathematics was 
coming into pretty clear consciousness. Plato himself is said to 
have been the inventor of the method. But in simple cases it is such 
a natural movement of thought that it is more than likely that it had 
already been employed vaguely and loosely by the mathematicians, 
and that Plato only brought it into full consciousness as a method. 



58 THE MATHEMATICAL ELEMENT IN PLATO'S PHILOSOPHY 

Then, having developed it in the field of an exact science — mathe- 
matics — he sought to apply it also in the field of philosophy. It 
will be convenient for our purpose to discriminate two phases in this 
method and to bring out the logical implications and significance of 
each of them. These two phases I shall call the positive phase and 
the negative phase. 

1. The positive phase- of analysis. — A certain proposition is up 
for investigation. It is assumed as true. Consequences are deduced 
on the basis of this assumption. If these consequences are known 
from some other source to be true, and if we can then start with 
them and reverse the process with the result of getting back to the 
original proposition, that proposition is proved to be true. If the 
original proposition cannot be thus deduced, nothing can be inferred 
either as to its truth or its falsity. Positive analysis with positive 
outcome is thus often inconclusive, and usually so in problems 
involving considerable complexity. If the conclusion, however, is 
known to be false, especially if it is in direct contradiction with some 
known truth or of accepted data (and hence is absurd), we have a 
right to infer that the original proposition is false ; for by correct 
processes of reasoning we cannot deduce from a true proposition 
consequences which are false. Thus positive analysis, whether with 
positive or with negative outcome, admits of no inference as to the 
truth or falsity of any proposition other than the original one from 
which we start, and for that proposition is successful most frequently 
only in proving its falsity. It is not, then, by itself of much use in the 
search for truth. 

2. The negative phase of analysis. — This is the complement of 
the positive phase, the two together forming a complete logical 
method. Here as before we have a proposition up for investigation. 
But we start with the assumption that it is not true — a proposition 
logically contradictory with the original one, or else with such an 
alternative proposition that the original proposition and it are 
mutually exclusive. If, in reasoning from the assumption, we arrive 
at a conclusion known to be false — i. e., in contradiction to known 
truths or of positions granted (and which is thus absurd) — then we 
have to reject the supposition which led to this conclusion, as in (1). 
But this proposition is so related to the original one that the rejection 
of it as false carries with it necessarily the acceptance of the truth 
of the original. Here, then, negative outcome of one proposition has 
with reference to a related proposition a positive significance. This 



METHOD, OR THE TECHNIQUE OF INVESTIGATION 59 

procedure is then a very powerful instrument of investigation in all 
cases where two mutually exclusive propositions can be stated, in 
which the truth of either involves the falsity of the other and the 
falsity of either involves the truth of the other. The method may 
be extended, of course, so as to apply to cases of three or more 
alternatives where the proved falsity of all but one, through analysis 
to contradictory or absurd conclusions, is proof of the truth of that 
remaining one. 

The above, I think, is a fair account of what is involved in the 
method of analysis, whether applied in mathematics or outside of 
mathematics. There would naturally be more variations in the 
details of its application in outside fields. These variations come in 
especially in the matter of starting the analysis, not with the propo- 
sition under discussion, but with other accepted truths. These are 
analyzed and conclusions drawn which are at variance with the 
original proposition. The inference, then, is that there is something 
wrong with that proposition. 

SOCRATIC ANALYSIS. 

There is no reason to suppose that Socrates was under the influ- 
ence of mathematical thought. Yet by discussing his method with 
reference to the outline given above we can get a clearer conception 
of the relation between his analysis and Platonic analysis, and 
especially of the differences between them. If we follow Jowett and 
Windelband, we may take as most representative of Socrates's own 
thought, and as least modified by the introduction of any reactive 
element on the part of Plato himself, the group of dialogues com- 
prising Charmides, Lysis, Laches, Protagoras, Euthyphro, Apology, 
and Crito, with the addition of Hippias Minor and Alcibiades I., 
which are classed as of doubtful genuineness. I shall omit the latter 
two from the discussion. However other scholars might arrange 
the Socratic dialogues, they would agree substantially on this group 
as fairly typical. 33 

The most uniform characteristic of this group of dialogues on 
the side of content is in their dealing with the problem of the virtues. 
This most frequently takes the form of an attempt at definition; 
e. g., Charmides, the definition of temperance ; Lysis, of friendship ; 

33 Joel and Shorey emphasize the Platonic element in them. My main argu- 
ment does not depend on either point of view. It would, however, harmonize 
better with that of Joel and Shorey. See p. 77. 



60 THE MATHEMATICAL ELEMENT IN PLATO'S PHILOSOPHY 

etc. Let us take them up and investigate them on the side of method. 
In Charmides five definitions of temperance are taken up in succes- 
sion. In every case there is a process of analysis to contradictions or 
absurdities. There is no case of negative analysis. There is no 
statement of true alternatives. There is one instance (160) which 
might at first be taken as such, where the statement is made : " And 
of two things, one is true — either," etc. But closer examination 
shows that it is an inadequate opposition of quietness and quickness. 
In the Lysis, again, it is positive analysis with negative outcome. 
There is a clear statement of the assumption on which such analysis 
rests : " If we had been right, we should never have gone so far 
wrong" (213). Some attempt is made at setting off alternatives, 34 
but they are not true alternatives and are used ambiguously in the 
course of the argument. In the Laches the analysis is all positive 
with negative outcome, and there is no explicit attempt to set off 
alternatives. The thought of the Protagoras moves in the form of 
positive analysis with negative outcome. There are no true alterna- 
tives. There is an appearance of the use of alternatives in several 
places in the setting of opposites against each other. In one place 
this effect is heightened by playing with the double meaning of the 
negative : if the virtues are distinct and separate, then " holiness is 
of the nature of the not-just, and therefore of the un-just, and the 
un-just is unholy ; " i. e., holiness is unholy. The opposites in 332-33 
are not used as alternatives, but for the purpose of showing that two 
opposites to the same thing are identical ; i. e., in the course of the 
argument temperance has been shown to be the opposite of folly, 
and wisdom has been shown to be the opposite of folly ; therefore 
temperance and wisdom are identical. In 359-60 recurs practi- 
cally the same thing with the addition of one step to the process. 
Courage has been shown to be the opposite of cowardice, wisdom the 
opposite of ignorance, and cowardice identical with ignorance ; there- 
fore courage and wisdom are identical. In the Euthyphro also we 
have the positive analysis with negative outcome. There is an 
apparent recognition of the principle of alternatives in the state- 
ment : " Then either we were wrong in our former assertion ; or if 
we were right then, we are wrong now." " One of the two must be 
true." But these alternatives are not used as pivots on which the 
analysis swings. They are given merely for the sake of emphasizing 
a contradiction, not for the sake of passing from it to a truth. One 

N Cf. Lysis, 216 and 222. 



METHOD, OR THE TECHNIQUE OF INVESTIGATION 6 1 

line of analysis, starting with the definition of piety as " that which is 
dear to the gods," has led to contradictions ; another line, starting 
with a different hypothesis, has led straight back to the definition 
again of piety as "that which is dear to the gods." This is con- 
fusion confounded, and the statement of the alternatives has the 
effect of heightening this confusion. The Apology and the Crito 
are more dogmatic and deductive in form, and from the nature of 
their subjects are hardly to be classed as dialogues of search. 

A common feature of the dialogues just discussed, and one 
inherent in their method, is their negative outcome — no satisfactory 
definition is found. The real argument starts with an accepted, or 
suggested, or tentative definition. This is analyzed, and from it con- 
sequences are deduced. These consequences are found to involve 
absurdities and contradictions. Or, if the start is made from other 
accepted truths, the consequences are found to be m contradiction 
with the definition or its consequences. Hence the suggested defini- 
tion is rejected, at least in its present form, and a new start has to 
be made with a new definition or some modification of the old. 

Socrates was a master of that kind of analysis which got at 
meanings empirically. He could also show the absurd consequences 
deducible from assumed data among loose thinkers, and thus it was 
that he was able to take the conceit out of the windbags of knowledge. 
The logical connection between the absurd consequences and the 
falsity of the original proposition he seems to have grasped. But he 
had not developed the method far enough to take advantage of its 
power in the derivation of new truth. The negative outcome of the 
Socratic dialogues is explained when we remember that they are 
cast in the form of positive analysis, and hence that logically the 
contradictory consequences do not allow of any inference to a nezv 
truth, but only to the falsity of the original proposition. The function 
of negative analysis and the necessity of working out from mutually 
exclusive alternatives have not yet been seen. There are, however, 
some gropings after the complete form of analysis. Opposites are 
frequently set off against one another, and alternatives are stated. 
But these are very loosely used, not being true opposities or mutually 
exclusive alternatives, or not being used in the forward and positive 
movement of thought. We do not mean to say that, apart from this, 
the Socratic analysis did not have a forward movement. But its 
movement toward the truth was by a series of hitches. A proposi- 
tion was analyzed to an absurd conclusion and rejected. A new 



02 THE MATHEMATICAL ELEMENT IN PLATO S PHILOSOPHY 

start was then made by choosing a new proposition or by modifying 
the original one in accordance with some suggestion received in the 
preceding process. This procedure might be kept up in some cases 
until some proposition was found the analysis of which did not neces- 
sitate as the outcome its rejection. But in this phase of the method it 
will be easily observed that there is no adequate control in the deter- 
mination of the direction in which to look for the truth. It was a 
sort of logical " cut and try " procedure, more or less empirical in 
character, and not truly scientific. 35 It cannot be determinate in the 
search for truth until negative analysis is introduced and mutually 
exclusive alternatives are worked out. Plato came to see this need 
and he devoted whole dialogues, so far as their method is concerned, 
to the exposition of it. 

ZENO'S ANALYSIS. 

Zeno's analysis also seems to be lacking in the element of intel- 
lectual control, although there are intimations that he was vaguely 
reaching out for some positive significance to the negative outcome 
of his analysis. There seems to be a vague feeling that, because the 
Heraclitean doctrine of motion leads to paradoxical conclusions, 
somehow the contradictory consequences of Eleaticism are mini- 
mized. In the Parmenides, Plato has Zeno say : 

The truth is, that these writings of mine were meant to protect the argu- 
ments of Parmenides against those who make fun of him and seek to show the 
many ridiculous and contradictory results which they suppose to follow from 
the affirmation of the one. My answer is addressed to the partisans of the 
many, whose attack I return with interest by retorting upon them that their 
hypothesis of the being of many, if carried out, appears to be still more 
ridiculous than the hypothesis of the being of one. 36 

Zeno's argument in no way shows, however, that the absurd con- 
clusions deducible from the Heraclitean hypothesis involve the truth 
of the Eleatic doctrine. Plato seems to feel that, while Zeno's analy- 
sis is significant, yet it is lacking in the element of intellectual con- 
trol. It is an easy matter to get all sorts of contradictions when you 
are operating in the realm of sense-perception. The Zenonian para- 
doxes lose their significance because they are not worked out rigor- 
ously and exclusively in the intellectual realm, the world of ideas. 37 

lor another statement of the value and the limitations of Socratic analysis, 
see WlNDELBAND, History of Ancient Philosophy, pp. 128-30. 
M Parmenides, 128. 37 Parmenides, 135. 



METHOD, OR THE TECHNIQUE OF INVESTIGATION 63 

PLATONIC ANALYSIS. 

The method of analysis can be found in all its phases and stages 
of development in the properly Platonic dialogues. It would be 
unnecessarily tedious to take up all the cases of its use, but I shall 
take up numerous illustrations in detail for the purpose of showing 
its significance clearly and making apparent the advance which it 
marks over the Socratic analysis. 

The Gorgias. — Ostensibly the subject of this dialogue is rhetoric. 
It opens with a presentation of Gorgias, the rhetorician, who praises 
his art. Socrates is drawn into an argument with him, and soon 
catches him in a contradiction. But the turn which the dialogue 
quickly takes leads one to think that Plato's real interest in writing 
the dialogue was the ethical one. There is an assumption on the 
part of Gorgias and his friends that rhetoric is a fine and useful art 
because of the advantage that it gives to a man to have the power 
of attaining his ends through persuasion, even though he may be 
inferior in knowledge. This view of rhetoric seems to Plato not 
alone superficial, but above all immoral ; for it makes the art the tool 
of the unrighteous against the righteous. He attacks the point of 
view of Gorgias by the assertion of a series of Socratic paradoxes. 
The real and vital argument of the dialogue centers about the fol- 
lowing paradoxical positions of Socrates; (1) that to do injustice 
is worse than to suffer it (469, 473, 474) ; (2) that when a man has 
done evil he is happier if he be punished than unpunished (473). As 
incidental to the proof of these there arise two subsidiary paradoxes : 
(a) bad men do what they think best, but not what they desire, for 
the desire of all is toward the good ; 38 (b) to be, and not to seem, is 
the end of life. 39 The position of his chief opponents, Polus and 
Callicles, may be summed up in the following statements : ( 1 ) 
might makes right (483-84) ; (2) law is nothing but the combination 
of the many weak against the strong (483) ; (3) the pleasant is the 
good (492). Now, while these two sets of propositions cannot 
literally be set off against each other as mutually exclusive alterna- 
tives, yet ethically considered the points of view which they represent 
are, taken as a whole, felt by the participants to be antagonistic and 
non-inclusive. In accordance with this feeling, the method employed 
in the defense and establishment of the Socratic paradoxes is indirect. 

38 Gorg., 466, cf. 509, end. 

39 Gorg., 511 to end of dialogue. See Jowett, Introduction to the Gorgias, 
2 : 270 ff. and 303 ff. 



64 THE MATHEMATICAL ELEMENT IN PLATO'S PHILOSOPHY 

The opposite point of view is attacked and shown to be shot through 
with contradictions when its implications and consequences are 
drawn out. Thus the negative of the Socratic position is proved 
untenable. The outcome of this negative conclusion of the negative 
analysis is of positive significance in establishing the truth of the 
Socratic position. The evidence for my interpretation of the method 
in which the Socratic paradoxes were proved is not alone internal, 
but also the explicit statement at the close of the argument upon 
them. The statement is as follows : 

These truths, which have been already set forth as I stated them in the 
previous discussion, would seem now to have been fixed and riveted by us, 
if I may use an expression which is certainly bold, in words which are like 
bonds of iron and adamant; and unless you or some other still more enter- 
prising hero shall break them, there is no possibility of denying what I say. 
For my position has always been, that I myself am ignorant how these things 
are, but that I have never met anyone who could say otherwise, any more 
than you can, and not appear ridiculous. 40 

The establishment of the ethical paradoxes of Socrates has a 
connection with the ostensible subject of the dialogue, namely the 
function of rhetoric. The pointing out of this connection is a device 
which preserves the literary symmetry of the work. At the same 
time, it is done in such a way as to be an integral part of the method 
of analysis. When the truth of these paradoxes is accepted, the 
much boasted usefulness of rhetoric vanishes on any supposition of 
its advantage in helping to attain unjustly one's ends ; for nothing 
can be useful except in so far as it helps one to do justly, or is used 
by him to persuade to his own punishment when he does wrong 
(480). Again, unless rhetoric is not to be a true art, but only a sort 
of flattery (462-67), the only alternative (implied) is the "position, 
which, acording to Polus, Gorgias admitted out of modesty, that he 
who would truly be a rhetorician ought to be just and have a knowl- 
edge of justice. 41 

Within the total movement of thought, several interesting 
instances of the use of the method of analysis might be pointed out. 
In the passage 492-95, Callicles maintains the identity of pleasure 
and the good, a position which Socrates attacks by showing the 
disagreeable consequences which follow from it, clinching his argu- 
ment with an explicit statement in 495 : " I would ask you to consider 
whether pleasure, from whatever source derived, is the good ; for 

m Gorg; 5<>'j- a Corg., 508. 



METHOD, OR THE TECHNIQUE OF INVESTIGATION 65 

if this be true, then the disagreeable consequences which have been 
darkly intimated must follow, and many others." Callicles still holds 
to his position, and Socrates attacks him through the mutually exclu- 
sive opposites of good and evil, identifying pleasure (on Callicles's 
hypothesis) with the good and pain with the evil: but pleasure and 
pain can coexist, then good and evil can coexist, which is contrary 
to the hypothesis that they are mutually exclusive opposites (495-97) . 
There is a still further analysis from another point of view in 497-99. 
In 475 there is a case of the explicit statement of alternatives, fol- 
lowed by the elimination of all but one, whereupon that one is 
regarded as proved true. Similarly in 477 and in 478. 

Here, then, within the limits of one dialogue are found both posi- 
tive and negative analysis, the use of alternatives implicit and 
explicit, and the attainment of positive conclusions — though there 
is some vagueness and lack of rigor in the use of the complete 
method. 

The Meno. — Early in the Meno virtue has been denned as " the 
desire of things honorable and the power of attaining them." Desire 
of the honorable is identified with desire of the good. Then begins 
an analysis of this definition. The very specification seems to imply 
that there are some who desire the evil also. This is admitted. 
Further analysis of this idea leads to the conclusion that they desire 
to be miserable and ill-fated, which cannot be held to be true. Then 
on this basis the definition has to be rejected. A return is then made 
to the definition, and it is attacked from a different point of view. 
The two parts of the definition are taken up separately: first as to 
the desire of things honorable, and secondly as to the power of 
attaining them. Analysis of the first leads to the unacceptable con- 
clusion that one man is no better than another in respect to virtue. 
Before analyzing the second, the qualification " with justice " is added 
to "the power of attaining them." But justice is a part of virtue, 
and we have the unsatisfactory conclusion of virtue defined in terms 
of a part of itself. On three different counts, then, Meno's definition 
of virtue has to be rejected (77-79). This reduction to absurd 
consequences from several points of view is quite characteristic of 
Plato's analysis. There are two illustrations of reductio ad 
dbsurdum in the famous geometrical demonstration with the slave 
boy. His answer that the side of the square of double the value of 
the square whose side is two feet will be double that of the given 
square is followed out so that he sees that such a square will have an 



66 THE MATHEMATICAL ELEMENT IN PLATO'S PHILOSOPHY 

area of sixteen square feet, whereas by hypothesis it ought to have 
only eight. Hence his answer is wrong and he must try again. His 
second answer that its side should be three feet is treated in like 
manner (82-83). 

In the cases given from the Meno thus far the analysis is positive 
with a negative outcome. No alternatives have been stated by 
means of which inference could be made to the truth of the other 
proposition on the basis of the falsity of the one examined. There 
are passages, however, where there is a working with alternatives. 
A new attempt is made to define virtue by identifying it with knowl- 
edge (87). If this is correct, then virtue can be taught. The diffi- 
culties of this position are analyzed out at some length, in the 
process of the analysis several subordinate points being made by 
working through alternatives (88, 89, 96). The contradictions to 
which the definition of virtue as knowledge leads calls for its rejec- 
tion. But there is an alternative proposition to this, namely, that 
virtue is right opinion ; for there are only these two guides to action 
— knowledge and right opinion (97, 99). This alternative proposition 
is regarded as true 42 by reason of the falsity of the other. This in 
turn has a bearing on the question whether virtue can be taught. 
Virtue is either natural, or acquired, or a God-given instinct 
(98-100). Whether it is right opinion or knowledge, it is not 
natural (98) ; if it is to be acquired, this must be because it is 
knowledge, a view which has already been rejected (98-99) ; virtue 
then is neither natural nor acquired, hence it must be what it is in 
order to be right opinion, a God-given instinct (99-100). 

The Euthydemus. — This dialogue appears to some trivial and 
meaningless. It is not so when one has firmly grasped the idea that 
Plato uses the method of analysis, not for the purpose merely of 
landing one in hopeless contradictions, as the eristics did, but as 
having some positive significance, even if that positive significance 
be not explicitly pointed out. The destruction of one point of view, 
with him, meant the acceptance of another. The Euthydemus is a 
satire of eristic, but it is more than that. It is an illustration of the 
absurd and contradictory consequences which can be drawn where 
definition is not careful and words are used ambiguously. This has 
its significance in teaching indirectly that the symbols of language 

42 That is, from the point of view of this discussion. Plato's own view seems 
to be that virtue in the highest sense is identical with wisdom in the highest sense. 
pp. 32-34. 89, 91 of this book. 



METHOD, OR THE TECHNIQUE OF INVESTIGATION 67 

are functional with reference to thought, and not necessarily fixed 
and unambiguous. Furthermore, the Euthydemus is a reductio ad 
absurdum of that view of judgment which gives the predicate an 
existential force or makes the judgment an identical proposition. 
The Republic. — In the first book of the Republic the discussion 
centers about the definition of justice. Cephalus defines justice " to 
speak the truth and pay your debts" (331). The first half of the 
definition is analyzed out to contradictory conclusion and abandoned. 
The second half is likewise analyzed out to conclusion which is 
absurd ; it is then remodeled, when again absurd conclusions are 
derived which make out justice to be useless. This results in still 
further modifications of the definition, which upon analysis again 
result in contradictions (331-36). Thrasymachus defines justice as 
"the interest of the stronger." This is reduced to contradiction 
with his own statement that it is just for the subjects to obey their 
rulers ; for the rulers may themselves err as to what is their interest 
(338-39). But Thrasymachus maintains that no artist or ruler qua 
artist or ruler is ever mistaken. In opposition to this it is then shown 
that the ruler in his capacity of ruler merely is interested in the wel- 
fare of his subjects — that is his sole business qua ruler. Justice then 
is their interest and not his, the interest of the weaker and not of the 
stronger — a conclusion which is contradictory to the original defini- 
tion which Thrasymachus proposed (340-42). Thrasymachus now, 
defeated in the argument, expounds at length the advantages of 
injustice (343-45). Put in the form of a proposition, his contention 
is that the life of the unjust is more advantageous than the life of 
the just; and, further, that injustice is virtue and wisdom, justice 
the opposite. Through an intermediate proposition which Thrasy- 
machus accepts the consequences are deduced that the just is wise 
and good; the unjust evil and ignorant (347-50). The second half 
of Thrasymachus's position has, then, to be rejected. Before taking 
up the first half, a little piece of negative analysis is introduced. 
Taking the conclusion just reached, the position can now be refuted 
that injustice is stronger and more powerful than justice ; for perfect 
injustice is shown to be self-destructive in its effects, defeating its 
own ends (351-52). Returning to the first half of Thrasymachus's 
position, through a doctrine of ends the conclusion is reached that 
justice is the excellence of the soul and injustice the defect; the just 
is happy and the unjust miserable. But happiness and not misery 



68 THE MATHEMATICAL ELEMENT IN PLATO'S PHILOSOPHY 

is profitable. Therefore injustice can never be more profitable than 
justice (352-54). 

It will now be seen that the essential movement of thought in 
the first book of the Republic is through positive analysis, unaccom- 
panied by alternative propositions. The outcome is negative, the 
position of Thrasymachus and his friends is overthrown; but no 
definition of justice is established in the place of those proposed. The 
definition of justice is reserved until its nature has been seen in the 
analysis of the ideal state. 

It may be noted that in the fourth book of the Republic there is a 
very clear case of the use of negative analysis. The alternatives are 
stated as follows : "Which is the more profitable, to be just and act 
justly and practice virtue, whether seen or unseen of gods and men ; 
or to be unjust and act unjustly, if only unpunished and uni- 
formed?" In the light of the previous discussion (all that follows 
Book I), the question is now declared to be absurd; for analysis of 
the second alternative shows that through injustice the very essence 
of the vital principle is undermined and corrupted, and under that 
condition it is inconceivable that life is worth the having. The first 
alternative, then, must be accepted. 

The Phcedo. — Some cases of the use of the method of analysis 
are found in the Phcedo. One is found in connection with the argu- 
ment for the pre-existence of the soul. Alternatives are worked out 
(75) and stated (76) : We come into life having knowledge; or 
knowledge is recollection. The first alternative is taken up for 
examination. If we come into life having knowledge, we ought to be 
able to give an account of it from the beginning, which we cannot do. 
The first alternative is then untrue, and the second is proved, namely, 
that knowledge is recollection. It is felt that this proof of the pre- 
existence of ideas carries with it the proof of the pre-existence of 
the soul (76, yy). But what about the soul's living after death? It 
is said that the soul is a harmony. Then just as the harmony dies 
with the perishing of the strings, so the soul passes away with the 
dissolution of the body. This argument is refuted from three differ- 
ent points of view in succession. (1) It is shown that this view of 
the soul leads to a conclusion which is contradictory to the pre- 
viously proved and accepted doctrine that knowledge is recollection 
(91-92). The conclusion to be drawn from this is clearly stated: 
"Having, as I am convinced, rightly accepted this conclusion [that 
knowledge is recollection], 1 must, as I suppose, cease to argue or 



METHOD, OR THE TECHNIQUE OF INVESTIGATION 69 

allow others to argue that the soul is a harmony" (92). (2) The 
assumption that the soul is a harmony leads to the conclusions : (a) 
of degrees in the being of the soul ; (b) of a harmony within a har- 
mony in case of the virtuous soul, and of an inharmony within a 
harmony in case of the vicious soul ; (c) all souls must be equally 
good. The significance of these curious and paradoxical conse- 
quences in refutation of the idea that the soul is a harmony is 
explicitly noted: "And can all this be true, think you? he said; 
for these are the consequences which seem to follow from the assump- 
tion that the soul is a harmony ?" (93-94.) (3) The assumption 
that the soul is a harmony involves the view that the soul cannot utter 
a note at variance with the tensions, relaxations, etc., of the strings 
of which it is composed. This is in contradiction with the known 
fact that the soul leads, opposes, and coerces the "elements" (94). 

From three different points of view it has now been proved by 
positive analysis with negative outcome that the soul is not a har- 
mony. But this is not the positive result desired, namely, that the 
soul is immortal and indestructible. This proof is led up to by a pre- 
liminary discussion which serves to secure a long series of accepted 
truths relative to the final argument. This series concludes with the 
deduction from the essentially opposite and mutually exclusive char- 
acter of life and death that the soul, which is the life of the body, 
cannot participate in death. This outcome is then made more rigor- 
ous by a further analysis both on the positive and the negative side. 
" If the immortal is also imperishable, then the soul will be imperish- 
able as well as immortal." But this positive analysis is not felt to be 
conclusive ; for if the soul is not immortal, " some other proof of her 
imperishableness will have to be given." But if the argument is put 
in the form of negative analysis, " no other proof is needed ; for if 
the immortal, being eternal, is liable to perish, then nothing is imper- 
ishable." This is contrary to fact in the case of God and the essential 
form of life. Therefore the soul is imperishable (100-107). 

The Theceietus. — The movement of thought in this dialogue is, 
taken as a whole, positive analysis with negative outcome. There 
are minor and subsidiary movements which might be otherwise 
classified as, e. g., a recognition of alternatives in some places (164, 
188, etc.), also at least one important doctrine developed by direct 
procedure ( 184-86) . The argument of the dialogue commences with 
an attempt on the part of Thesetetus to define knowledge. In the 
course of the dialogue three such attempts are made and discussed. 



JO THE MATHEMATICAL ELEMENT IN PLATO S PHILOSOPHY 

1. Knowledge is sense-perception (152-86). This is identified 
with the doctrine of relativity of the Protagorean school, and it is 
discussed largely from that point of view. But first of all the doc- 
trine of relativity is itself developed so as to show what it involves 
in its relation to this problem. It is interesting to note that in this 
ancillary portion of the task of refuting the definition the method of 
analysis is employed. Perception may be relative (a) to the subject 
and the object, the percipient and the perceived; (b) to the sub- 
ject and an object which is itself relative; (c) to a relative subject 
and a relative object, each having but a momentary existence. Each 
of these possible meanings of relativity is taken up in order and 
found to involve contradictions. The tacit assumption is, in each 
case, that when the cruder form of the doctrine of relativity breaks 
down through contradictions inherent in it there is an alternative, 
a way of escape, through taking refuge in a more refined form of the 
doctrine. In this way the doctrine is developed to its utmost logical 
limit. When this is done, it is found to involve difficulties still. But 
waiving these aside for the time being, a return is made to the defini- 
tion itself. The fundamental assumption of this definition is the 
identity of knowledge and sense-perception (163 fL). Analyzing 
this assumption, it is found to involve verbal contradiction (163-65). 
In connection with the assumption, "What seems to a man is to 
him," the doctrine of identity breaks down again (170-84), through 
analysis of it to a conflict with common-sense and other conflicts 
(170-71) and the destruction of any possibility of judgments involv- 
ing futurity (177-79). A return is then made to the doctrine of 
"universal flux," and it is found also to involve irreconcilable con- 
clusions. An examination is now made as to the sources of those 
elements of conscious experience which we are most ready to admit 
as knowledge, and it is found that they do not come through the 
sense-organs (184-86). This reconstruction and the negative out- 
come of the positive analysis both coincide in proving the falsity of 
the definition of knowledge as perception. 

2. Knowledge is true opinion (187 ff.). This definition is taken 
up and analyzed, the first thing being noted that the specification 
" true " opinion would seem to imply the existence of false opinion. 
When this assumption is examined, it is found that in the sphere of 
knowledge false opinion is impossible (187-88), and likewise in the 
sphere of being (188-89) 5 hence it must be sought elsewhere, if at 
all. There seems to be one other alternative — that false opinion is 



METHOD, OR THE TECHNIQUE OF INVESTIGATION Jl 

a sort of heterodoxy, a confusion of one thing for another (189). A 
list of cases is drawn up where such confusion is impossible, and 
these are then excluded from consideration (192). The only remain- 
ing possibility is the confusion of thought and sense (193). Is it 
true ? A serious difficulty arises from its failure to explain mistakes 
about pure conceptions of thought, like numbers (196). The out- 
come is that "we are obliged to say, either that false opinion does 
not exist, or that a man may not know that which he knows." The 
former alternative seems to be the only one possible. A further 
analysis of knowledge reveals the fact that the accounting for false 
opinion is bound up with the problem of defining knowledge. Hence 
a return is made to the original question, and the examination is 
resumed of the definition of knowledge as true opinion (200). But 
in the law court the lawyer may judge rightly on the basis of true 
opinion without knowledge. Now, " if true opinion in law courts 
and knowledge are the same, the perfect judge could not have judged 
rightly without knowledge ; " for knowledge and true opinion are by 
hypothesis identical. But he did give the right judgment without 
knowledge, and " therefore I must infer that they are not the same " 
(201). This final argument is almost a perfect reductio ad absurdum 
of the identity of knowledge and perception. A new attempt has to 
be made. 

3. Knowledge is true opinion combined with reason or explana- 
tion (201-10). This definition is attacked in the same way. If 
explanation means pointing out the elements of a compound, no gain 
is made by the addition of the term to the definition of knowledge ; 
for analysis reveals insuperable difficulties. Giving a reason may 
mean reflection of thought in speech, enumeration of the parts of a 
thing, or a true opinion about a thing with the addition of a mark or 
sign of difference. In either of the first two senses, contradictions 
are deducible; and in the third sense you finally get knowledge 
defined in terms of itself, which is not a definition at all. The third 
definition of knowledge has then failed like the other two. 

The final outcome of the Thecetetus is negative. It could not well 
be otherwise when cast in the form of positive analysis. The 
definitions discussed are not related in an alternative or mutually 
exclusive way; hence there is no opportunity to infer from the 
proved falsity of two of them to the truth of the third. Yet this 
negative outcome has some positive significance in the mind of 
Plato. In the Parmenides the problem of being and not-being is 



*]2 THE MATHEMATICAL ELEMENT IN PLATO S PHILOSOPHY 

discussed at some length, and the difficulties of both conceptions are 
exploited. In the Sophist this problem of not-being is conceived of 
as at bottom one with the problem of false opinion. Without going 
into the discussion in detail, it might be well to point out the con- 
clusion reached there. 

If not-being has no part in the proposition, then all things must be true; 
but if not-being has a part, then false opinion and false speech are possible, 
for to think or to say what is not — is falsehood, which thus arises in the 
region of thought and in speech. 43 

In undermining a theory of generalization like that of the modern 
associational school of Locke and Mill, 44 and like it based on an 
associational psychology, and in showing the inadequacy of the 
existential conception of judgment, Plato prepared the way for the 
further analysis and reconstruction of the function of judgment and 
of the negative which is worked out in the Parmenides and the 
Sophist. I hold that it is a mistake to suppose that Plato was neces- 
sarily ignorant of the bearing of the negative outcome of the 
Thecetetus, or of any other dialogue, merely because he defers dis- 
cussion of the problem till some other time. It certainly is a remark- 
able fact how he makes use of such negative outcomes in further 
reconstructions along positive lines. 

The Parmenides. — It has already been noted that this dialogue 
may be regarded from one point of view as a long and thorough 
exposition of the method of analysis (see p. 50). Here first the 
positive and negative phases of analysis receive explicit and specific 
recognition as necessary parts of one complete method of investiga- 
tion. This statement is worthy of quotation. 

But I think that you should go a step further, and consider not only the 
consequences which flow from a given hypothesis, but also the consequences 
which flow from denying the hypothesis. (136.) 

As an illustration of what is meant by this procedure, the Par- 
menidean hypothesis of the one is taken up and examined from every 
point of view on both the positive assumption and on the negative. 
The larger part of the dialogue is taken up with this analysis. It is 
preceded, however, by a critique of the Platonic Ideas. 

The most apparent division of the dialogue is into two parts: 
(i) a criticism of Platonic Ideas, (2) a criticism of the Eleatic doc- 
trine of Being. I think, however, that the real function of the 

'Sophist, 260; cf. 261, beginning. u Thecel., 201 ff. 



METHOD, OR THE TECHNIQUE OF INVESTIGATION 73 

dialogue is somewhat different from that which appears on the sur- 
face from an observation of subject-matter. The result of the first 
investigation seems at first to be a proof of the untenability of the 
Ideas. The hypothesis is shown to involve great difficulties. There 
is the problem of the relation of individuals to the Ideas. Is it one of 
participation or of resemblance ? Then, too, the process of referring 
back to an Idea, when once started, would seem to have to go on to 
infinity. And, thirdly, there is the difficulty of the relation of the 
ideas within us to absolute Ideas. Yet, in spite of these difficulties, 
Plato feels that the doctrine of Ideas is not to be abandoned. There 
is an alternative, the consequences of which are far more disastrous 
than those deducible from the doctrine of Ideas. That alternative is 
the non-existence of these Ideas. He feels that there are difficulties 
in the other position, but that this is wholly untenable, necessitating 
the acceptance of the other in spite of its difficulties. This is wholly 
in keeping with the movement of Plato's thought and his method of 
procedure. The way it is put in the Parmenides is as follows : 

And yet .... if a man, fixing his attention on these and the like difficul- 
ties, does away with Ideas of things and will not admit that every individual 
thing has its own determinate Idea which is always one and the same, he will 
have nothing on which his mind can rest; and so he will utterly destroy the 
power of reasoning.* 5 

The criticism of the Eleatic doctrine of Being seems not to have 
its greatest significance in the outcome with reference to that problem, 
but in its bearing upon the function of the copula and the negative 
in judgment. The eristics had made predication impossible, 46 through 
their treatment of the judgment as existential. Also the negative 
" is not " was given the existential force and made to signify abso- 
lute non-existence. 47 The judgment, then, if positive, could be 
nothing but an identical proposition, and hence valueless ; if negative, 
was an absurdity and impossibility. 

There is both in Greek and in English an ambiguity in the mean- 
ing of " is." The eristics played upon this ambiguity in such a way 
as to throw the emphasis wholly upon the existential force of the 
word, and thus brought out their contradictions of ordinary common- 
sense. Plato out-eristics the eristics in weaving to and fro between 

45 Parmen., 135 ; cf. Soph., 259-260, 249. 

48 Soph., 251 E, 259 E, 251 C; Theat., 201 E-202 A. These references from 
Shorey's Unity of Plato's Thought, p. 58, footnote 433. 

47 Soph., 238 C-241 A ; Shorey, op. cit., footnote 434. 



74 THE MATHEMATICAL ELEMENT IN PLATO S PHILOSOPHY 

the two meanings of the copula. He shows himself by his analysis a 
master not only of their game of producing contradictions, but also 
goes them one better by analyzing their own position to contradictory 
conclusions. All this, it seems to me, is something more than a play 
or a satire. It is a bringing to clear consciousness the fact that there 
is an ambiguity in the use of the copula and of the negative. When 
this is seen, the judgment can become a vital knowledge-process, 
having a function denied to it when viewed solely in the existential 
sense. In the negative judgment also you have not merely an asser- 
tion of not-Being. In the very denying of one thing to the subject 
you virtually assert otherness of the subject; in saying that a 
thing is not this, you are not saying that it is not anything, but that 
it is other than this. The only not-Being that is intelligible to 
thought is such not-Being as is implied in otherness. 48 

The analysis both positive and negative of the Parmenides, 
though it results in both cases in a negative outcome through the 
contradictions which are reached, is a preparation for a reconstruc- 
tion of the signification of predication and of negation. The negative 
outcome may be explained from the fact that, though we have the 
two phases of analysis here, yet they are not made to work through 
wholly unambiguous terms. It is another point of significance to 
this dialogue that it shows so clearly the necessity of viewing lan- 
guage as something other than a static thing, and hence in arguing it 
is necessary to use the terms employed always in the same sense. 
The abstract and highly rational use of such terms as "one," 
" being," " other," " like," " same," " whole," and their opposites is 
a different thing from their concrete use. 49 As concrete terms any 
sort of conclusion can be deduced from them through playing upon 
variations in their meaning. 50 

The Sophist. — The argument of the Sophist is in large part in 
the form of the method of analysis. The problem of not-Being, 
where the term is used in the sense of absolute denial of existence 
and absolute separation from Being, is taken up in this way. The 
contradictions involved in predication, and even in the mere use of 
the word itself, are pointed out. The inference from this is made 
that the assumption is false, and that Parmenides's philosophy must 
be put to the test. Plato undertakes to show that such a separation 
between Being and not-Being must be abandoned ; and he explicitly 

** See Shorey, op. cit., pp. 58, 59. 

m Cf, Phileb., 14-15. w Parmen., 135 ; cf. Soph., 259. 



METHOD, OR THE TECHNIQUE OF INVESTIGATION 75 

points out his reason for thinking so — the unavoidable contradic- 
tions which result from the Parmenidean position (237-41). His 
rejection of the various forms of philosophy, which he examines at 
some length (242-51) with reference to this problem, is on the 
ground of the contradictions into which they fall when analyzed out. 
The inference from these negative results of a supposed separation 
of Being and not-Being is that they ought not to be separated abso- 
lutely. But this does not mean that they necessarily mingle abso- 
lutely. And here comes in one of the best illustrations of analysis 
through alternatives. There are three possible alternatives : (1) no 
participation, (2) indiscriminate participation, (3) participation or 
intercommunion of some ideas with some. Each of these is taken 
up in turn. The first two are rejected on the ground of their con- 
tradictory consequences ; and the third is accepted as the only 
remaining alternative. The whole argument is followed by a care- 
ful summary so that the full positive force of the reductio ad 
absurdum is brought out (251-52). Having established this doc- 
trine of the intercommunion of ideas, he proceeds to develop it and 
to apply it to the reconciliation of the contradictions previously 
deduced in the Sophist and also in the Parmenides (253-58), indi- 
cating explicitly that one source of such contradictions, as was 
pointed out in our discussion of the Parmenides, is the verbal shift- 
ing of words and meanings (259). He concludes his argument on 
this point of the separation of Being and not-Being by an argument 
against the universal separation of classes that is very characteristic 
of the way in which he is always going back to the principle of con- 
tradiction and making it yield positive results rather than merely 
negative ones. This is his statement: 

The attempt at universal separation is the final annihilation of all reason- 
ing; for only by the union of conceptions with one another do we attain to 
discourse of reason. 51 

Any proposition that leads to the annihilation of reasoning or the 
impossibility of knowledge has been reduced to an absurdity and has 
to be abandoned. Having disposed of the absolute separation of 
Being and not-Being by a general argument, and thus made possible 
the reconciliation of the contradictions of the Parmenides, he pro- 
ceeds, as has already been shown (see p. 72), to apply the conception 
of the nature of not-Being just reached to the solution of the problem 
of false opinion in the The at etas. Thus the Sophist is the develop- 

51 Soph., 259 end to 260 beginning; cf. Parmen., 135; Soph., 249. 



y6 THE MATHEMATICAL ELEMENT IN PLATO'S PHILOSOPHY 

ment of the positive significance of the negative outcome of both the 
Thecetetus and the Parmenides. Predication is again made possible 
and significant. The copula and the negative in the judgment have 
significance in the knowledge process. 

The Statesman and the Sophist. — We have taken up but one 
phase of the Sophist. The other phase can be discussed in connec- 
tion with the Statesman. Both these dialogues aim to get at defini- 
tions through the process of logical division. The definitions of the 
Sophist and of the Statesman come at the end of a long process of 
dividing species with the greatest care in the matter of getting classes 
that are mutually exclusive, until at last the thing sought to be 
defined is caught in a final class in such a way as to be distinguished 
from all other things and at the same time to have its own essential 
nature indicated. The Ideas, as was shown in the Sophist, have 
intercommunion some with others. Hence the problem of definition 
is the problem of dividing them off properly, while at the same time 
preserving their integrity as to the principle that runs through the 
whole. 

He who can divide rightly is able to use clearly one form pervading a 
scattered multitude, and many different forms contained under one higher 
form; and again one form knit together into a single whole and pervading 
many such wholes, and many forms existing only in separation and isolation. 52 

Summary. — The study of the foregoing dialogues is a revelation 
of the fact that Plato was familiar with and used the method of 
analysis in all its phases. In one dialogue one phase may be pre- 
dominant, in another another, according to the purpose to be con- 
served. In some places the main object is the destructive one of 
clearing away obstacles to the position that he wishes to maintain. 
No positive conclusion is cared for ; the main thing is refutation. 
Here positive analysis, with its negative outcome, is wholly adequate ; 
and it is not necessary to suppose that this negative outcome has, in 
the mind of Plato, no positive significance. Positive analysis is also 
adequate when the main object is to satirize the position of his 
opponents or contemporaries, or when he skilfully stimulates the 
curiosity and awakens the interest of his hearers by leading them 
into a tangle of contradictions with reference to things which they 
thought that they understood perfectly. But when he wishes to 
secure positive results, he also knows how to set off alternative propo- 

'- Soph., 253. 



METHOD, OR THE TECHNIQUE OF INVESTIGATION JJ 

sitions against each other, either of which excludes or negatives 
the other, so that by proving one of them false he lias the right to 
infer from this negative outcome positively to the truth of the other 
proposition. The advance over what I have called Socratic analysis, 
whether that really represents Socrates's method, or whether it is 
employed purposely by Plato himself in that group of dialogues 
merely because it was adequate to the purpose in mind, 53 is in the 
use of negative analysis, especially in that form in which alternatives 
are either clearly stated or are clearly in mind. The Thecetetus is a 
good illustration of positive analysis taken by itself ; the Parmenides, 
of both positive and of negative analysis in more or less isolation 
from each other, so far as the inference to new truth is concerned; 
the Sophist and the Statesman exhibit the method whereby mutually 
exclusive alternatives may be derived ; the Sophist furnishes a good 
illustration of the power of analysis when conscious use is made of 
the leverage which is given by mutually exclusive alternatives. Such 
instances may be found elsewhere, with a greater or less degree of 
perfection. So markedly do the phases, and the results of the differ- 
ent phases, of analysis stand out in the Thecetetus, the Parmenides, 
and the Sophist and Statesman taken together, that one might with 
some reason argue that they were written with the pedagogic pur- 
pose in mind of exhibiting the method of analysis in detail. 

THE RELATION OF MATHEMATICS TO PLATONIC ANALYSIS. 

The clearest positive intimation of the influence of mathematics 
in the determination of Plato to the use of the method of analysis 
is to be found in the Meno. There the suggestion is made to discuss 
the question of whether virtue can be taught by assuming a hypo- 
thesis and deducing consequences, as in geometry. 54 " Now, this argu- 
ment from hypothesis, as we have seen at some length, is very char- 
acteristic of Plato's procedure. This he himself recognizes explicitly 
in several places, aside from the internal evidence which we have 
given. 55 

Philosophical problems usually involve great complexity. On 
this account, while we may admit that it is possible that all the logical 
steps involved in the method of analysis might have been discovered 
wholly within the field of philosophical discussion, yet this is improb- 
able. Especially is this true when the same method is actually being 

53 See footnote, p. 59 of this book. " Meno, 86-87 ; see pp. 84 ff. of this book. 
55 See Phcedo, 99-100 ; Parm.,136; Gorg., 509 ; Phcedo,io6; Rep., 6 : 510-11. 



y8 THE MATHEMATICAL ELEMENT IN PLATO'S PHILOSOPHY 

used in mathematics — a field of investigation where the intellectual 
control of problems can be made more perfect, where relations are 
more sharply defined, and where simplicity is attainable in the 
highest degree. Now, Plato was interested in pure mathematics, and 
he was especially interested in mathematics on the side of its qualities, 
characteristics, processes, methods, and, in general, everything that 
had any philosophical or logical significance. Whether Plato's inter- 
est in the method of analysis had its origin on the side of philosophy 
or of mathematics makes little difference. When once this interest 
had dawned, it would find its greatest opportunity of realizing itself 
in complete logical form within the field of mathematics. It is also 
characteristic of Plato to study method in easier and clearer cases 
first and then to apply it to the more difficult. 56 W T e might naturally 
expect that he would first come to clear consciousness of this method 
in mathematics. In doing so, as we seem justified in inferring that 
he did, and as tradition confirms, he at once made a distinguished 
contribution to the logic of mathematics and at the same time got 
the clue to the essential conditions that the method must fulfil in order 
to be of service as a rigorous instrument of investigation in philoso- 
phy. It was under the influence of the mathematical element that 
he got the stimulus which made him transcend Socratic and Zenonian 
analysis by the introduction of those phases of the method which 
make it complete. 

58 See Rep., 2 : 368-69 and Soph., 218 ; discussion on p. 48 of this book. 



CHAPTER IV. 

RELATION OF MATHEMATICAL PROCEDURE TO DIALECTIC. 

It would appear from the preceding discussion that mathematical 
procedure — at least the method of analysis in some form — is on the 
logical side the most fundamental feature of Plato's dialogues of 
investigation. The term " dialectic " is quite loosely used to signify 
in general any procedure which gets at a new truth or higher point of 
view through discussion and analysis. In this sense of the word it 
includes Socratic analysis and also mathematical analysis. In some 
places Plato's use of the term "dialectic" makes it a sort of poetry. 
The soul gazes directly upon the reality of the universe, beholds 
unfettered by sense the eternal being by the aid of pure intelligence 
alone, 1 and finds in so doing her true love ; here dialectic is akin to 
love, 2 a feeling of affinity with the truth. In this sense of the word, 
dialectic would include the mathematical process in so far as direct 
intuition is involved. But there are many places where Plato uses 
the term "dialectic" in a more restricted and technical sense, and 
where he appears, at least, to make a distinction between dialectic and 
mathematical procedure. This makes necessary some discussion of 
the relation of mathematical procedure to dialectic proper. 

As has been suggested before, Plato seems to have been subject 
to a twofold movement of thought, the activities of which ran 
parallel to each other, interacting upon and modifying each other. 
One phase of his thinking moved along the path of a logical a priori 
demand ; the other was mathematical. The first movement was 
closely connected with a fundamental interest of his — namely, the 
practical, or ethical. When the validity of ethical standards was 
impeached by the Protagorean sensationalistic philosophy, Plato 
dreamed of a method which should secure results free from skeptical 
outcome by being empirical in none of its elements or processes. It 
should attain its conclusions solely through the exercise of the reason. 
Its data, its processes, its results, should all be rational. All the 
ethical concepts should be deducible from hypotheses, or principles, 
demanded by an active intelligence, not imposed by sense, and these 
in turn should be traced back step by step to one supreme teleological, 

1 Phcedrus, 247. 2 Symp., 210 ff.. 

79 



80 THE MATHEMATICAL ELEMENT IN PLATO'S PHILOSOPHY 

non-empirical principle — the Idea of the Good. Such a method 
Plato conceived would give us knowledge of true being — abiding, 
changeless, eternal. The problems which center in securing this are 
par excellence the problems which should engage the thought of the 
philosopher. The method which would thus work wholly in the 
realm of the rational and secure absolute knowledge is called by 
Plato dialectic. Dialectic is, then, in this more technical sense of the 
word, the ideal of philosophic investigation. It is the demand of 
the a priori logical movement of thought. But described in these 
terms it has as yet little specific content. This content will come out 
in further discussion. 

With reference to the points just made — Plato's ethical problem, 
and his feeling of the need of finding a method of attack which should 
proceed along wholly rational lines 1 — a passage in the Phcedo 
(96-101) is very significant. Here Plato seems to have reached the 
point where he is unwilling to accept the statement of conditions as 
an explanation of any phenomenon of nature or fact in mathematics, 
but he demands an explanation in terms of final cause — a teleological 
explanation. How to give such an explanation is his problem. He 
feels that the key to its solution is to be found in rational rather than 
in natural process. He " has in mind," he says " some confused 
notion of a new method." 3 so when he finds Anaxagoras saying that 
" mind is the disposer and cause of all," 4 he hails this notion with 
" delight," thinking that at last he is going to have the solution of 
his problem. The ground of his hope in Anaxagoras was that he 
thought that when he spoke of mind as the disposer of all things, 
he would show how all things are as they are because this was best. 5 
He expected to see cause identified with the good. He then goes on 
to tell of his great disappointment in Anaxagoras, for he learned only 
of conditions and not at all of final causes. The futility of such 
explanations he illustrates by supposing that the reason why Socrates 
sits and awaits his execution instead of running away be given in 
terms of the structure and function of the various parts of the body, 
instead of in terms of his " choice of the better and nobler part." In 
developing this point, he says : 

There is surely a strange confusion of causes and conditions in all this. 
It may be said, indeed, that without bones and muscles and the other parts 

3 Phaedo, 97; dXXd tip'' &X\ov rpbtrov avrbs eUji <f>6pa). 

* vovs i(mv 6 diaKocr/Auiv re kclI ir&VTUV alrtos. 

11 c5 (X €LV i /SAtkttos, and dfjieivuv are used. 



RELATION OF MATHEMATICAL PROCEDURE TO DIALECTIC 8 1 

of the body I cannot execute my purposes. But to say that I do as I do 
because of them, and that is the way in which mind acts, and not from the 
choice of the best, is a very careless and idle mode of speaking. I wonder 
that they cannot distinguish the cause from the condition. 6 

Again, in explaining the relations of the physical universe they 
make the same blunder of ignoring final cause. 

Any power which in arranging them as they are arranges them for the best 
never enters into their minds, and instead of finding any superior strength in 
it, they rather expect to discover another Atlas of the world who is stronger 
and more everlasting and more containing than the Good (rb dyadbu) ; of 
the obligatory and containing power of the Good they think nothing, and 
yet this is the principle (rrjs rotavnjs alrias') I would fain learn, if anyone 
would teach me. 7 

Thus both in the realm of conduct and in the realm of nature 
Plato is seeking for explanation in terms of final cause ; and without 
question in the realm of ethics he identifies that cause with the prin- 
ciple of the Good. Such is the outcome of the ethical problem for 
Plato when he follows along the path of the logical a priori demand 
— a demand which, we have seen, itself sprang out of a reaction 
against a particular solution of the ethical problem. He has come to 
the distinction between sense and intellect ; and, throwing stress upon 
rational process, this emphasis being in turn strengthened by the 
mathematical influence, he has exalted mind to the highest place. 
But mind, intelligence, presupposes purpose. Ethics demands that 
this purpose be in the direction of the Good. Thus he reaches the 
demand for the teleology of the Good. In the Idea of the Good we 
have united both the rational, which is necessary in order to transcend 
the doctrine of relativity, and also the ethical. Plato feels that this 
is the outcome that is required. But what the method, or technique, 
of obtaining it? Certainly not any that admits empirical elements at 
any stage. 8 In the Phcedo he " has in mind some confused notion of 
a new method." 

We have seen the conditions out of which the demand came for a 
new method, and also the conditions which this new method, dialectic, 
must fulfil — what its nature in general must be. How is such a 
method to be evolved? Dialectic in this technical sense must cer- 
tainly be a long and tedious process — the elimination of the sense- 
elements well-nigh impossible. He himself indicates that only after 
■•■he severest practice can dialectic be mastered. It involves the 

* Phcedo, 99. 7 Phcedo, 99. 8 Cf. Phileb., 58, 59, 61. 



82 THE MATHEMATICAL ELEMENT IN PLATO'S PHILOSOPHY 

severest abstraction and the most highly rational processes. Also 
it involves elements of direct intuition on the part of a highly devel- 
oped and exceedingly active and keen mental sight. Its processes 
cannot be readily exhibited, any more than the process of seeing 
green with the physical eye can be exhibited and explained to one 
who has never seen. Hence Plato turns to mathematics — the second 
path along which his thought moved to the same goal. Here is a 
process, which, in one realm — that of a particular exact science — 
attains to ideas wholly rational and free from the sense-element. 
This process serves as the model — the ideal which should be attained 
in every realm of philosophic thought. Mathematical method gives 
the cue to Plato for working out the problem of the " new method " 
which his definition of the ethical problem demands. Whatever may 
have been the difficulties in the mathematical method of reaching the 
goal of rational conclusions, it had for Plato the very great advantage 
of being actually capable of having its processes exhibited. Further- 
more, it had so much in common with the method of which he was 
in search that training in it served as direct mental preparation for 
the exercise of dialectic. 

As dialectic came to the full and clear consciousness of Plato in 
its relation to mathematical procedure, the nature of dialectic as a 
process can be best explained by a more detailed discussion of the 
relation of mathematical procedure to dialectic. What Plato seized 
upon as most suggestive in mathematical procedure was the method 
of analysis with its hypothetical procedure. 9 This has already been 
explained at length. This method served as the point of departure 
for him in the formulation of his dialectic method. Also there was a 
style of argumentation prevalent against which Plato reacted. This 
he called eristic. The nature of dialectic needs to be studied in rela- 
tion to this as well as in relation to the mathematical method of 
analysis, both in order to understand dialectic better and also in 
order to understand the significance of the mathematical element in 
dialectic. 

Eristic starts from premises, but diflers radically from dialectic 
in spirit. In the first place, the eristic prefers to start from his own 
premises, which he considers true unless his opponent can refute 
them ; the dialectician is willing to start from the premises of his 
opponent and analyze them out to their conclusion. If he starts with 
his own premises, it is always with the assent of his opponent. 10 

Meno, 86-87; Rep., 6:510-11. 10 Meno, 75, 79; Soph., 259. 



RELATION OF MATHEMATICAL PROCEDURE TO DIALECTIC 83 

Now, what does this difference imply ? It means that the dialectician 
is fully conscious of the hypothetical character of his inquiry, while 
the eristic is not. The dialectician is interested in the interrelation of 
premises and conclusions, realizing that a concluson of a certain sort 
has as much significance with reference to the premises as that 
premises of a certain sort have with reference to the conclusion, and 
he wants to sift out the truth in so far as that is a matter of the 
relation of premises and conclusion. Plato is fully conscious of the 
hypothetical character of his method and explicitly recognizes it. 11 
In the second place, as has just been intimated, the dialectician 
approaches a discussion in the spirit of one who is searching for 
truth; the eristic, as one who will maintain a point, more particu- 
larly as one who delights to puzzle and overwhelm his opponent with 
contradictions. This is brought out in several passages. I will give 
one or two citations. 

The disputer (x«ty>is 5t diaXeydfievos) may trip up his opponent as often as 
he likes, and make fun; but the dialectician will be in earnest, and only cor- 
rect his adversary when necessary. 12 

He will imitate the dialectician who is seeking for truth, and not the 
eristic, who is contradicting for the sake of amusement. 13 

In the third place, the eristic (avTiAoyiKoY) confuses the hypothesis 
and its consequences 14 — a point closely related to the preceding, 
whereas the dialectician understands their true relation to each other. 
The eristic has a tendency to take as final his conclusions, or at least 
to leave the discussion without any help for those who have been 
following it with reference to a postive outcome. He rejoices in 
having left them in the midst of puzzles and contradictions which 
seem hopeless. Indeed, that is the aim of the whole argument ; for 
it gives everyone the impression of superior argumentative power on 
his part. The dialectician, on the other hand, while he may lead 
up to just as absurd, paradoxical, and contradictory conclusions, yet 
does so with a consciousness of the fact that these conclusions are so 
bound up with the premises that in coming out as he has done he has 
a right to a further inference, namely, with reference to the truth or 
falsity of the premises. He uses negative outcomes, not as neces- 
sarily final, but as indices of the need of reconstruction or of further 
inference. Dialectic is more than an instrument of refutation ; it is a 
process of investigation. 

lx Ph<zdo, 106, 107. 13 Rep., 7:539; cf. 5 : 454 A ; 6 : 499 A ; Soph., 259. 
12 Thecetetus, 167 E. 14 Phatdo, 101 E. 



84 THE MATHEMATICAL ELEMENT IN PLATO'S PHILOSOPHY 

The nature of dialectic can be seen with still greater clearness and 
with more of technical force by studying the relation between mathe- 
matical procedure and dialectic. The position has already been taken 
in the discussion of Plato's divided line that, so far as their ideal 
character is concerned, there is no difference between mathematical 
concepts and the Ideas. The third and fourth divisions on the side of 
content were held to be not essentially different, but it was intimated 
that there was an essential difference on the side of process. The 
point which I wish to make here is that this difference on the side 
of process is precisely the difference between mathematical and 
dialectical method ; and that such difference as may be felt on the 
side of content as the result of this is not so much one of essence as 
of cognitive aspect. While I would agree with Milhaud in part, yet 
it seems inconceivable that Plato should have dwelt so much on the 
difference between mathematics and dialectic, if there was no sense 
in which the distinction was in his mind significant. With this pre- 
liminary statement of position, I will now turn to the discussion in 
more detail, so as to make this point clearer. 

However high a value Plato may set upon the work of the 
mathematician, it is still true " that the skilled mathematician is not a 
dialectician." 15 In this same context he intimates the ground upon 
which he makes this difference. It is that the mathematician cannot, 
like the dialectician, give a reason (\6yov). In what sense this is true 
will be seen later. I give the statement here to bring out vividly 
the fact that Plato does not indentify mathematical procedure and 
dialectic. He seems to feel that the mathematician is under more or 
less constraint from the sense-element. His initiative is not found 
in a free activity of thought. There is something given to him 
behind which he does not go. " Mathematicians do not make their 
diagrams, but only find out that which was previously contained in 
them.'' 16 They start with data that are given, or at the best from 
some fixed point intellectually determined, and reason through an 
intuition of relations and logical connections to necessary conclusions. 
But dialectic examines the validity of the data themselves. Plato brings 
this point out very clearly in his discussion of the different use 
which mathematics and dialectic make of hypotheses. Before further 
discussion, it may be well to give some significant citations. 

The inquiry [in the third division, where mathematics has been placed] 
can only be hypothetical, and instead of going upward to a principle (^r' 

13 Rep., 7: 531 D, E. 10 Euthydemus, 290. 



RELATION OF MATHEMATICAL PROCEDURE TO DIALECTIC 85 

apxnv) descends to the other end; in [dialectic or SiaXe/criK^] , the soul passes 
out of hypotheses (inrodfoewv) and goes up to a principle (apx?iv) which 
is above hypotheses {awirbderov), making no use of images (eUbwav) as in the for- 
mer case, but proceeding only in and through the ideas themselves (avroTs 
etdeiri). 1 ? 

You are aware that students of geometry, arithmetic, and the kindred 
sciences assume (inro0£p.evoi) the odd and the even and the figures and three 
kinds of angles and the like and their several branches of science: these 
are their hypotheses (u7ro06reis) which they and everybody are supposed 
to know, and therefore they do not deign to give any account of them either 
to themselves or others; but they begin with them and go on until they 
arrive at last, and in a consistent manner at their conclusion. 18 

And when I speak of the other division of the intelligible (rod vorjrod,) 
you will understand me to speak of that other sort of knowledge which 
reason (6 \670s) herself attains by the power of dialectic (rod duiXiyetrdai dwdp.ec), 
using the hypotheses (faodfoeii), not as first principles (apx&s), but only 
as hypotheses (r£ 5ptc vTrodtcreis), that is to say, as steps and points 
of departure into a world which is above hypotheses (avvirSderov), in order 
that she may soar beyond them to the first principles of the whole (tt]v rod 
iravrbs apxyv), and, clinging to this, and then to that which depends on this, 
by successive steps she descends again without the aid of any sensible object 
(at<rdrrr£) from ideas, through ideas, and in ideas she ends (etdevip airois 5i' 
avruv ct's atird, ical reXevrq. els efSTj). 19 

A careful study of these passages reveals both likenesses 
and differences between mathematical process (Stavota) and dialectic 
(vo*7<xis). Both make use of hypotheses. Mathematical process 
(Siavoia) does not discuss their validity; dialectic (vo^o-it) traces 
them back to their ground, not regarding them as self-sufficient. 
Both seek ideas free from sense. Mathematical process (Siavoia) 
uses visible symbols as a means ; dialectic (vorjo-is) makes no use of 
sensuous symbols. But the significant thing is the difference in the 
use of hypotheses. In the one case of the use of the method of 
analysis the mathematicians seem to have come to a somewhat clear 
conception of the hypothetical character of their reasoning. They 
are not aware that their ordinary data are also hypothetical in char- 
acter, that they rest back upon something else. They view them as 
apxat not as viroOivtis. The dialectician goes to the full length of 
seeing, not alone the data of the method of analysis as hypothetical, 
but also the whole body of mathematical data. It would seem that, 
starting from the one clear instance of the hypothetical method in 

17 Rep., 6: 510 B. 

18 Rep., 6:510 C, D. 19 Rep., 6:snB; cf. Phoedo, 101 D, E. 



86 THE MATHEMATICAL ELEMENT IN PLATO'S PHILOSOPHY 

mathematics, Plato perfected the method of analysis in this realm, 
and, then using it as a model of investigation, there grew up the 
conception of dialectic, in which the same principle of investigation 
is carried over to all fundamental concepts. The method is univer- 
salized, and that, too, in two directions : ( I ) in the extension of its 
field of application beyond the realm of mathematics ; (2) in its 
depth of application, in the demand that all the fundamental concepts, 
not alone of mathematics, but also of all subjects, be subject to 
examination with reference to the possibility of grounding them in 
some higher principle. When this principle has been reached, then a 
descent is possible through wholly intellectual and rational process. 
Dialectic involves both the ascending and the descending processes. 
Thus a system is formed within which the free activity of the mind 
may exercise itself without being under the constraint of sense- 
perception, yet in accordance with the principles and laws of reason. 
Such a system would meet the demands imposed by following out 
the logical a priori line of thought, and would at the same time 
involve all that Plato had found of value in the mathematical method 
of procedure. 

This same conception of dialectic as the universalization of the 
hypothetical method comes out in the Sophist (253). All the char- 
acteristics of the method of analysis are there hinted at, only univer- 
salized. There is the process of division, whereby, as we have seen, 
mutually exclusive alternatives are secured. There is the search to 
see if any universal class is dependent upon some other which makes 
it possible (hypothetical procedure). There is the ascending pro- 
cess — the attempt to see many different forms contained under one 
higher form ; and the descending process — the viewing of one form 
as knit together into a single whole and pervading many such 
wholes. This passage seems worthy of quotation in its entirety. 

And wiil he [the dialectician] not ask if the connecting links are universal, 
and so capable of intermixture with all things ; and again, in divisions, 
whether there are not other universal classes, which make them possible? 
.... Should we not say that the division according to classes, which neither 
makes the same other, nor makes other the same, is the business of dialectical 
science? That is what we should say. Then, surely, he who can divide 
rightly is able to see clearly one form pervading a scattered multitude, and 
many different forms contained under one higher form; and again, one form 
knit together into a single whole and pervading many such wholes, and many 
forms, existing only in separation and isolation. 20 

90 Soph., 253. 



RELATION OF MATHEMATICAL PROCEDURE TO DIALECTIC 87 

From this point of view, dialectic is not concerned with one art 
any more than with another. Its function is universal. All arts are 
on an equality before it — that of the general and that of the vermin 
destroyer. 21 It is concerned only with their pretensions or claims as 
arts. Its " endeavor is to know what is and what is not kindred in 
all arts, with a view to the acquisition of intelligence. 22 

Now, we may say that it is because dialectic is this intelligent 
aspect of all the arts that it is able to look upon them in this impar- 
tial manner as to their degree of honor or dishonor. Dialectic sees 
them all in their relation to the whole, as dependent upon some one 
common fundamental principle, on their relation to which their 
function and value depends. 

This brings us back to the problem of the relation of mathe- 
matical procedure to dialectic from a new point of view. Mathe- 
matical procedure has transcended the productive arts by bringing 
out from one point of view what is common to them all, common not 
merely as an element, but as a principle of control, something on 
which they all alike depend, namely, the principle of measure or 
quantity. But still, from Plato's point of view, the mathematical 
principle is not the highest point of view from which to see all things. 
Mathematics as such just misses possessing the characteristic of the 
highest kind of knowledge ; it stops short with its fundamental con- 
cepts unexamined, and hence does not have the comprehensive, all- 
inclusive, view of the whole which dialectic has. It itself needs 
further interpretation in the light of some further and higher prin- 
ciple. From this point of view it belongs in a realm somewhere 
between opinion (8o£a) and reason (vovs). Plato places it in the realm 
of the understanding (Siavota). 23 

If we work this point out still further, I think that it will give 
further light upon the relation which Plato sets up between mathe- 
matical procedure and dialectic. In the Thecetetus there is quite a 
discussion of opinion (Sd£a). There a distinction seems to be set up 
between opinion (So£a) and true opinion (So£a aXrjOrjs) , 24 It is not 
denied that the judge may decide rightly on the basis of true opinion 
(B6ia aXrjOrjs). In outcome the result is precisely the same as if he 
had knowledge and could give a reason. But yet his judgment is 
lacking in that cognitive aspect which entitles it to be called knowl- 

21 Soph., 227 B. 23 Rep., 6 : 51 1 D. 

22 Soph., 227; cf.. Farm., 130 D, E. 2i See especially Theaet., 201 B. C. 



88 THE MATHEMATICAL ELEMENT IN PLATO'S PHILOSOPHY 

edge in the higher sense. 25 So it is with mathematics, Plato feels. 
The mathematician does not use his results, but turns them over to 
the dialectician. 26 If Plato had used the terminology of true opinion 
and of the higher knowledge, he would have placed mathematics under 
true opinion. He does place it between opinion and reason, and 
that is where true opinion would have to fall. This interpretation 
would reconcile the apparent ambiguity, if not contradiction, in his 
stress upon the idealistic and absolute character of the results of 
mathematics, while at the same time making dialectic superior. In 
true opinion the content of mathematics, the essences, would be Ideas ; 
but, viewed from the side of process, these mathematical Ideas would 
be lacking in the complete cognitive aspect of knowledge in the 
higher, more philosophical, sense. This interpretation, while making 
intelligible the difference which Plato makes between mathematics 
and dialectic, also makes intelligible the close relations which he is 
continually pointing out and insisting upon. In the Meno (98), 
while insisting in the most vigorous language that knowledge differs 
from true opinion, he yet shows that true opinion may pass over into 
knowledge " when fastened by the tie of cause." This fastening is 
there done by recollection (in the metaphysical sense). We have 
already pointed out that Plato in many places uses dialectic in a sort 
of poetical fashion. It is no stretch of the facts as we know them in 
the Republic and the Phcedo to say that this fastening of true opinion- 
down by a " chain " with " the tie of cause," which is the function of 
recollection, is identical with the process of dialectic in other places. 
The particular bearing of this point on the problem in hand is found 
in a statement in the Republic (511) in immediate connection with 
the discussion of the use of hypotheses in mathematics and in dia- 
lectic. There, in speaking of mathematical truths as inferior to those 
of dialectic, the very significant qualifying clause is added/' although 
when a first principle is added to them they are cognizable by the 
higher reason." This statement, it will be observed, is in exact har- 
mony with the one in the Meno regarding true opinion. 

As the outcome of this interpretation we cannot say offhand and 
without qualification that Plato identifies, or that he distinguishes 
mathematics and dialectic in the knowledge-function. In content, 
viewed alone, they are alike ; in attitude of mind they differ in 
cognitive aspect ; in process they differ in degree and universality in 
the use of the hypothetical method. With reference specifically to 

■ Cf. Gurg., 454-55- " u Euthyd., 290. 



RELATION OF MATHEMATICAL PROCEDURE TO DIALECTIC 89 

the point of the influence of mathematical method upon Plato's con- 
ception of dialectic, the outcome of the discussion seems to show 
that there is good reason to believe that, while there were other 
factors at work to determine the movement of Plato's thought, yet 
the suggestion which he received from the application of the hypo- 
thetical method in mathematics was a very real factor in clarifying 
his thought and enabling him to formulate a method that should 
satisfy the logical demands under which he felt obliged to think his 
way through the ethical problem. 

We have seen how pressure was continually brought to bear upon 
Plato to set up distinctions of value on the cognitive side. The solu- 
tion of the ethical problem, from the logical a priori point of view, 
demanded the setting up of a distinction between the senses and the 
intellect. Mathematical procedure made the same demand. The 
distinction was set up. The analogy of the arts also conduced to the 
same end through its revelation of the principle of intellectual control 
involved in the mathematical element. But mathematics was not ade- 
quate to a knowledge of ends, nor was its method of procedure such 
as to be wholly adequate in giving knowledge which could be main- 
tained from every point of view equally well — knowledge which 
could always be carried back to a further hypothesis, until it rested 
on that which was above hypothesis. So a distinction had to be set 
up between mathematics and dialectic, and dialectic had to be defined 
from this further point of view and to meet this further demand. 
Dialectic thus became the upper limit of knowledge. In this dialectic 
we have the height of the rational activity, the mind freed from the 
constraint of that necessity which is involved in the sense-perception 
type of experience. The ascent has been made to a principle intel- 
lectually determined, and the descent can be made by intellectual 
processes to all the particulars which fall under the control of that 
principle. The ideal is realized of procedure from ideas to ideas by 
means of ideas. Herein Plato finds the possibility of the solution of 
the epistemological problem, and together with it the solution of the 
ethical problem (stated on p. 79) out of the severe demands of which 
the cognitive problem grew. Ethical principles no longer need to 
rest upon any Protagorean doctrine of relativity. The senses may be 
inadequate; let that be granted. But when virtue is based on 
knowledge, and knowledge comes through the process of dialectic, 
then the fundamental ethical principles rest upon a secure basis — a 
basis not subject to the law of becoming and change. 



90 THE MATHEMATICAL ELEMENT IN PLATO S PHILOSOPHY 

Now that we have followed through the process of setting up dis- 
tinctions, and have seen how Plato used these distinctions to state 
the epistemological problem in a way to ground ethics upon firm 
theoretic principles, it may be well to raise the further question as to 
the nature of these distinctions : Are they absolute or functional ? 
When the highest distinction has been worked out, has it any rela- 
tion of significance with reference to those which are lower? And 
have the lower any such relation of significance with reference to the 
higher ? There certainly is much to be said in defense of the thesis 
that the distinctions which Plato sets up are functional rather than 
absolute. 

The relation between the lower and the higher here is in many 
respects analogous to the relation between means and end. Mathe- 
matics came in at one level to mediate, to serve as the instrument of 
intellectual control, or organization of means with reference to 
results ; dialectic performs the mediating function at a higher intel- 
lectual level still. 

Even the senses have their value, but, from the full cognitive 
point of view, not as standing alone. They are inadequate rather 
than unnecessary. Their results have to be turned over to some 
other faculty to be judged, tested, and used. As is brought out in 
the Laws, the mind must mingle with the senses in order to secure the 
salvation of all. The safety of the ship is dependent, not on the pilot 
alone, nor on the sailors alone. It is when the perceptions of both 
pilot and sailors are united with the piloting mind that the ship is 
saved, together with those upon it. 27 In the city of the Laws the 
younger guardians have " their souls all full of eyes, with which they 
look about the whole city. They keep watch and hand over their 
perceptions to the memory, and inform the elders of all that happens 
in the city." These elders are wise in council, " and making use of 
the younger men as their ministers, and advising with them — in this 
way both together truly preserve the whole state." 28 In such 
imagery as this Plato indicates that there is a relation of connection 
between the distinctions which he has set up. From the side of value 
the higher is in a position of independence, while yet it is functionally 
related to the lower. 

This point of the relation between the distinctions of higher and 
lower is well illustrated in the cases already cited (see p. 88) of true 
opinion becoming knowledge when " fastened by the tie of cause," 

27 Laws, 12:961. ™ Laws, 12: 964-65. 



RELATION OF MATHEMATICAL PROCEDURE TO DIALECTIC 9 1 

and of mathematical truths " cognizable by the higher reason when a 
first principle is added to them." There is much also in the myth of 
the cave, or den, in the Republic, which would be in harmony with 
this same point of view. 

This attitude of Plato with reference to the relation of the highest 
kind of knowledge, that of the philosopher, or wise man, to the lower 
and less critical forms of knowledge, is carried right over, as we 
might expect, into the realm of ethics. This is seen most clearly by 
starting from the side of the analogy of the arts. The individual arts 
have some legitimate place, function, and value in their isolation; 
but the knowledge they involve is not knowledge in the highest 
sense, because it is not exercised with reference to a further end or 
good than the mere making of some article as well as it can be made. 
The wise man, the philosopher, views the excellence of the art, not 
alone on the side of the adequacy of the technique to produce a cer- 
tain product, but also with reference to the relation of that product 
to some ultimate and final end. He, then, has the higher knowledge 
with reference to this art. So it is with all the conduct of life. Habit, 
custom, routine in the realm of moral conduct, give virtues which are 
blind. These virtues become intelligent and truly ethical only when 
seen in relation to the Idea of the Good — the highest dialectical 
principle in the realm of ethical judgment. 

With the attainment of dialectic we have seen that Plato was able 
to solve his epistemological, and particularly his ethical, problem. 
The ontological problem, already partly swallowed up in the idealistic 
point of view (see pp. 29, 43) received its final solution in the identifi- 
cation of the Ideas, realized in the procedure of dialectic, and the 
highest and most ultimate reality, or essence — immaterial and eternal 
— in relation to which the whole world of the things of experience 
is to be judged. 

The cosmological problem also receives its solution. We have 
seen that mathematics came in to furnish the technique of production, 
and hence of the creation of the cosmos. But such technique implies 
intelligence, implies purpose, implies ultimate ends. These can be 
realized only through dialectic. Cause gets stated in teleological 
terms. As in the arts there was a workman who utilized the tech- 
nique which rested on mathematical principles, so in the cosmological 
process we find the demiurge, the embodiment of this rational activity 
of mathematics. In the arts there are premonitions of ultimate goods, 
but in the cosmological process, from the point of view of dialectic, 



92 THE MATHEMATICAL ELEMENT IN PLATO'S PHILOSOPHY 

there is clearly and intelligently grasped an ultimate end, the Idea of 
the Good, which the creator, seeks to realize. 

While the main problem of this book has been the significance 
of the mathematical element in Plato's philosophy, the discussion of 
that element could not be adequate except in the setting and context 
of the problems with which it was connected. Hence it seemed neces- 
sary to give in a more or less dogmatic way a statement of the solu- 
tion of these problems. However, the significance of the mathematical 
element does not come in so much in the solution of the problems as 
in the matter of its influence upon their formulation and the leverage 
which it gave to the problem of working out a method which Plato 
could regard as adequate to the process of investigation. The cap- 
sheaf of his method was dialectic, and for his conception of dialectic 
he owes very much to his interest in mathematical procedure. 



BIBLIOGRAPHY. 

ON THE HISTORY OF GREEK MATHEMATICS. 

Cantor, Moritz. Vorlesungen uber die Geschichte der Mathematik. Zweite 
Auflage. Leipzig: Teubner, 1894. Pp. 1-222. 

Hankel, Hermann. Zur Geschichte der Mathematik in Alterthum und 
Mittelalter. Leipzig: Teubner, 1874. Pp. 1-88. 

Suter, Heinrich. Geschichte der mathematischen Wissenschaften. Erster 
Theil, zweite Auflage. Zurich, 1873. Pp. 1-50. 9 

Bretschneider. Die Geometrie und die Geometer vor Euklides. Leipzig, 
1870. 

(Ahmes.) Ein mathematisches Handbuch der alien Aegypter (Papyrus 
Rhind des British Museum), iibersetzt und erklart von Dr. August 
Eisenlohr. Leipzig: J. C. Hinrichs, 1877. 

Chasles, M. Afiercu historique sur Vorigine, etc., de geometrie. Paris, 1837, 
1875. 

Allman, George Johnston. Greek Geometry from Thales to Euclid. Lon- 
don : Longmans, Green & Co., 1889. 

Gow, James. A Short History of Greek Mathematics. Cambridge: Uni- 
versity Press, 1884. 

Cajori, Florian. History of Mathematics. New York : Macmillan Co., 1894. 

ON THE PHILOSOPHICAL PROBLEM. 

General works on the history of philosophy are not included in 
this list. Likewise I have omitted particular reference to the volu- 
minous general Platonic literature. My work has been based most 
largely upon an original study of the dialogues. Hence I shall give 
only a brief list of works most closely related to my special problem : 
Cohen, Dr. Hermann. Platonic Ideenlehre und die Mathematik. Marburg, 

1879. 
Joel, Karl. "Der \6yos EtoKparucfc/' Archiv fiir Geschichte der Philosophic 

Bd. VIII (1895), PP. 466, 896; Bd. IX, p. 50. 
Benn, Alfred. "The Idea of Nature in Plato." Archiv fiir Geschichte der 

Philosophic Bd. IX (1896), p. 24. 
Milhaud, G. Les philosophes geometres de la Grece. Paris, 1900. 
Windelband, Wilhelm. Platon. (In Fromann's "Klassiker der Philoso- 
phic" ) Dritte Auflage. Stuttgart, 1900. 
Rodier, G. "Les mathematiques et la dialectique dans le systeme de Platon," 

Archiv fur Geschichte der Philosophic, Bd. XV (1902), p. 479. 
Ritchie, David G. Plato. New York : Scribner's, 1902. 
Shorey, Paul. 1'he Unity of Plato's Thought. ("University of Chicago 

Decennial Publications.") Chicago: University of Chicago Press, 1903. 

93 



INDEX. 

See also Table of Contents and Table of References to Passages in Plato 
involving Mathematics. This index does not include references to names of the 
Dialogues. For these see footnotes. The words " Plato " and " Mathematics " 
occur so frequently that nothing is gained by indexing them. 
Ahmes papyrus, 18. 

Analogy of the arts, 23-24, 28 ff., 32-34, 89, 91. See Arts. 
Analysis, 12, 13, 19, 57 ff., 63 ff., 82. 
Anaxagoras, 38, 80-81. 

Arithmetic, 9, 10, 12, 13, 14, 15, 16, 27-28, 36, 41, 44, 85. 
Arts, mathematics necessary to, 10, 12, 14, 15-17, 20, 28, 34-36, 42, cf. 87. See 

Analogy of arts. 
Astronomy, 17, 19, 20, 34~35, 39, 42-43- 
Contradiction, principle of, 51-52, 75. See Analysis. 
Cosmology, 22, 34, 37-40, 91. 
Definition, 10, 11, 18-19, 55 - 57> 66. 
Descartes, 51. 
Dialectic, 32, 44, 57, 79-92. 

Distinction between sense-perception and knowledge, 24-28, 30-31, 40-46, 53, 81. 
Divided line, 44-46, 84. 

Eleatics and Eleaticism, 21-22, 24-25, 29, 34, 37, 62, 72, 73, 74-75. 
Epistemology, 24, 28, 29-32, 34, 45-46, 54, 87-91. See Knowledge and Relativity of 

Knowledge. 
Eristic, 66, 73, 82-83. 

Ethics, 19, 23-25, 28, 32-34, 53, 63, 67, 79-81, 89-91. See Virtue. 
Euclid, 17, 19. 
Eudemian summary, 18. 

Geometry, 10, 11, 14, 19, 36, 41, 42, 44, 55, 77, 85. See Solid Geometry. 
Good, Idea of, 80-81, 91-92. 
Harmony, 17, 19, 20, 34~35, 4^-43, 44- 

Heracliteans and Heracliteanism, 21-22, 24, 25, 29, 31, 34, 37, 62. 
Hume, 51. 

Hypothesis and hypothetical method, 12, 19, 77, 79, 82-86. See Analysis. 
Idealism, 28, 34, 40-46, 54, 91. 
Ideas, 45-46, 72 ff., 76, 80, 91. See Good. 
Incommensurables, 16. 
Judgment, 55, 72-74. 75~76. 
Kant, 51, 55. 
Knowledge, not relative, 25-26, 27, 28, 30-32, 41 ff., 45, 50-52, 55, 69-72, 87-89. 

See Epistemology and Relativity of Knowledge. 
Locke, 51, 72. 
Mathematical games, 13. 
Milhaud, 45, 51, 55, 84. 

95 



96 THE MATHEMATICAL ELEMENT IN PLATO'S PHILOSOPHY 

Mill, 72. 

Music, 35. See Harmony. 

Mysticism and mathematics, 16, 20, 40. 

Number theory, Pythagorean, 7. 

Ontology, 22, 29, 38. 

Opinion, 31, 41, 45, 75, 87-90. See True opinion. 

Pedagogy, 13, 47~50. 

Philosophy, mathematics necessary to, 10, 43-44, 81-82. 

Protagoras and Protagoreanism, 22-23, 24, 27, 31, 53, 70, 79, 89. 

Pythagoras and Pythagoreanism, 7-8, 16, 17, 18, 43. 

Reason, 45-46, 86, 88. See Divided line. 

Recollection, doctrine of, 10-11, 55, 68. 

Relativity of Knowledge, 22-25, 69-72. See Epistemology and Knowledge. 

Sciences, mathematics necessary to, 9, 12, 15-17, 20, 28, 34-36, 42. 

Shorey, 22, 59, 73, 74- 

Slave boy, in Meno, 19, 47, 65. 

Socrates, 19, 23, 24, 28, 30, 32, 49, 52-57, 59-62, 77-79- 

Socratic universals, 53-54. 

Solid Geometry, 9, 17, 19, 39. See Geometry. 

Sophists, 22, 33, 48. 

True Opinion, 31, 50, 66, 70-71, 87-89. See Opinion and Knowledge. 

Understanding, 45-46, 87-89. See Divided line. 

Universals, Socratic. See Socratic universals. 

Virtue, in relation to knowledge, 12, 23-24, 28 ff., 32-33, 66, 77, 79, 89, 91. 

Zeno, 62, 78. 



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